author | wenzelm |
Thu, 25 Jul 2024 10:30:22 +0200 | |
changeset 80616 | 94703573e0af |
parent 78750 | f229090cb8a3 |
permissions | -rw-r--r-- |
78202 | 1 |
(* Title: HOL/Analysis/Urysohn.thy |
78127
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2 |
Authors: LC Paulson, based on material from HOL Light |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3 |
*) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
4 |
|
78202 | 5 |
section \<open>The Urysohn lemma, its consequences and other advanced material about metric spaces\<close> |
78127
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
6 |
|
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
7 |
theory Urysohn |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
8 |
imports Abstract_Topological_Spaces Abstract_Metric_Spaces Infinite_Sum Arcwise_Connected |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
9 |
begin |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
10 |
|
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
11 |
subsection \<open>Urysohn lemma and Tietze's theorem\<close> |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
12 |
|
78274
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
13 |
proposition Urysohn_lemma: |
78127
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
14 |
fixes a b :: real |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
15 |
assumes "normal_space X" "closedin X S" "closedin X T" "disjnt S T" "a \<le> b" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
16 |
obtains f where "continuous_map X (top_of_set {a..b}) f" "f ` S \<subseteq> {a}" "f ` T \<subseteq> {b}" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
17 |
proof - |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
18 |
obtain U where "openin X U" "S \<subseteq> U" "X closure_of U \<subseteq> topspace X - T" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
19 |
using assms unfolding normal_space_alt disjnt_def |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
20 |
by (metis Diff_mono Un_Diff_Int closedin_def subset_eq sup_bot_right) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
21 |
have "\<exists>G :: real \<Rightarrow> 'a set. G 0 = U \<and> G 1 = topspace X - T \<and> |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
22 |
(\<forall>x \<in> dyadics \<inter> {0..1}. \<forall>y \<in> dyadics \<inter> {0..1}. x < y \<longrightarrow> openin X (G x) \<and> openin X (G y) \<and> X closure_of (G x) \<subseteq> G y)" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
23 |
proof (rule recursion_on_dyadic_fractions) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
24 |
show "openin X U \<and> openin X (topspace X - T) \<and> X closure_of U \<subseteq> topspace X - T" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
25 |
using \<open>X closure_of U \<subseteq> topspace X - T\<close> \<open>openin X U\<close> \<open>closedin X T\<close> by blast |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
26 |
show "\<exists>z. (openin X x \<and> openin X z \<and> X closure_of x \<subseteq> z) \<and> openin X z \<and> openin X y \<and> X closure_of z \<subseteq> y" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
27 |
if "openin X x \<and> openin X y \<and> X closure_of x \<subseteq> y" for x y |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
28 |
by (meson that closedin_closure_of normal_space_alt \<open>normal_space X\<close>) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
29 |
show "openin X x \<and> openin X z \<and> X closure_of x \<subseteq> z" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
30 |
if "openin X x \<and> openin X y \<and> X closure_of x \<subseteq> y" and "openin X y \<and> openin X z \<and> X closure_of y \<subseteq> z" for x y z |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
31 |
by (meson that closure_of_subset openin_subset subset_trans) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
32 |
qed |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
33 |
then obtain G :: "real \<Rightarrow> 'a set" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
34 |
where G0: "G 0 = U" and G1: "G 1 = topspace X - T" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
35 |
and G: "\<And>x y. \<lbrakk>x \<in> dyadics; y \<in> dyadics; 0 \<le> x; x < y; y \<le> 1\<rbrakk> |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
36 |
\<Longrightarrow> openin X (G x) \<and> openin X (G y) \<and> X closure_of (G x) \<subseteq> G y" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
37 |
by (smt (verit, del_insts) Int_iff atLeastAtMost_iff) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
38 |
define f where "f \<equiv> \<lambda>x. Inf(insert 1 {r. r \<in> dyadics \<inter> {0..1} \<and> x \<in> G r})" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
39 |
have f_ge: "f x \<ge> 0" if "x \<in> topspace X" for x |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
40 |
unfolding f_def by (force intro: cInf_greatest) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
41 |
moreover have f_le1: "f x \<le> 1" if "x \<in> topspace X" for x |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
42 |
proof - |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
43 |
have "bdd_below {r \<in> dyadics \<inter> {0..1}. x \<in> G r}" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
44 |
by (auto simp: bdd_below_def) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
45 |
then show ?thesis |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
46 |
by (auto simp: f_def cInf_lower) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
47 |
qed |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
48 |
ultimately have fim: "f ` topspace X \<subseteq> {0..1}" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
49 |
by (auto simp: f_def) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
50 |
have 0: "0 \<in> dyadics \<inter> {0..1::real}" and 1: "1 \<in> dyadics \<inter> {0..1::real}" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
51 |
by (force simp: dyadics_def)+ |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
52 |
then have opeG: "openin X (G r)" if "r \<in> dyadics \<inter> {0..1}" for r |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
53 |
using G G0 \<open>openin X U\<close> less_eq_real_def that by auto |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
54 |
have "x \<in> G 0" if "x \<in> S" for x |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
55 |
using G0 \<open>S \<subseteq> U\<close> that by blast |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
56 |
with 0 have fimS: "f ` S \<subseteq> {0}" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
57 |
unfolding f_def by (force intro!: cInf_eq_minimum) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
58 |
have False if "r \<in> dyadics" "0 \<le> r" "r < 1" "x \<in> G r" "x \<in> T" for r x |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
59 |
using G [of r 1] 1 |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
60 |
by (smt (verit, best) DiffD2 G1 Int_iff closure_of_subset inf.orderE openin_subset that) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
61 |
then have "r\<ge>1" if "r \<in> dyadics" "0 \<le> r" "r \<le> 1" "x \<in> G r" "x \<in> T" for r x |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
62 |
using linorder_not_le that by blast |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
63 |
then have fimT: "f ` T \<subseteq> {1}" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
64 |
unfolding f_def by (force intro!: cInf_eq_minimum) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
65 |
have fle1: "f z \<le> 1" for z |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
66 |
by (force simp: f_def intro: cInf_lower) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
67 |
have fle: "f z \<le> x" if "x \<in> dyadics \<inter> {0..1}" "z \<in> G x" for z x |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
68 |
using that by (force simp: f_def intro: cInf_lower) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
69 |
have *: "b \<le> f z" if "b \<le> 1" "\<And>x. \<lbrakk>x \<in> dyadics \<inter> {0..1}; z \<in> G x\<rbrakk> \<Longrightarrow> b \<le> x" for z b |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
70 |
using that by (force simp: f_def intro: cInf_greatest) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
71 |
have **: "r \<le> f x" if r: "r \<in> dyadics \<inter> {0..1}" "x \<notin> G r" for r x |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
72 |
proof (rule *) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
73 |
show "r \<le> s" if "s \<in> dyadics \<inter> {0..1}" and "x \<in> G s" for s :: real |
78202 | 74 |
using that r G [of s r] by (force simp: dest: closure_of_subset openin_subset) |
78127
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
75 |
qed (use that in force) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
76 |
|
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
77 |
have "\<exists>U. openin X U \<and> x \<in> U \<and> (\<forall>y \<in> U. \<bar>f y - f x\<bar> < \<epsilon>)" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
78 |
if "x \<in> topspace X" and "0 < \<epsilon>" for x \<epsilon> |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
79 |
proof - |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
80 |
have A: "\<exists>r. r \<in> dyadics \<inter> {0..1} \<and> r < y \<and> \<bar>r - y\<bar> < d" if "0 < y" "y \<le> 1" "0 < d" for y d::real |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
81 |
proof - |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
82 |
obtain n q r |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
83 |
where "of_nat q / 2^n < y" "y < of_nat r / 2^n" "\<bar>q / 2^n - r / 2^n \<bar> < d" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
84 |
by (smt (verit, del_insts) padic_rational_approximation_straddle_pos \<open>0 < d\<close> \<open>0 < y\<close>) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
85 |
then show ?thesis |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
86 |
unfolding dyadics_def |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
87 |
using divide_eq_0_iff that(2) by fastforce |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
88 |
qed |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
89 |
have B: "\<exists>r. r \<in> dyadics \<inter> {0..1} \<and> y < r \<and> \<bar>r - y\<bar> < d" if "0 \<le> y" "y < 1" "0 < d" for y d::real |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
90 |
proof - |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
91 |
obtain n q r |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
92 |
where "of_nat q / 2^n \<le> y" "y < of_nat r / 2^n" "\<bar>q / 2^n - r / 2^n \<bar> < d" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
93 |
using padic_rational_approximation_straddle_pos_le |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
94 |
by (smt (verit, del_insts) \<open>0 < d\<close> \<open>0 \<le> y\<close>) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
95 |
then show ?thesis |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
96 |
apply (clarsimp simp: dyadics_def) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
97 |
using divide_eq_0_iff \<open>y < 1\<close> |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
98 |
by (smt (verit) divide_nonneg_nonneg divide_self of_nat_0_le_iff of_nat_1 power_0 zero_le_power) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
99 |
qed |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
100 |
show ?thesis |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
101 |
proof (cases "f x = 0") |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
102 |
case True |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
103 |
with B[of 0] obtain r where r: "r \<in> dyadics \<inter> {0..1}" "0 < r" "\<bar>r\<bar> < \<epsilon>/2" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
104 |
by (smt (verit) \<open>0 < \<epsilon>\<close> half_gt_zero) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
105 |
show ?thesis |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
106 |
proof (intro exI conjI) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
107 |
show "openin X (G r)" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
108 |
using opeG r(1) by blast |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
109 |
show "x \<in> G r" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
110 |
using True ** r by force |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
111 |
show "\<forall>y \<in> G r. \<bar>f y - f x\<bar> < \<epsilon>" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
112 |
using f_ge \<open>openin X (G r)\<close> fle openin_subset r by (fastforce simp: True) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
113 |
qed |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
114 |
next |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
115 |
case False |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
116 |
show ?thesis |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
117 |
proof (cases "f x = 1") |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
118 |
case True |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
119 |
with A[of 1] obtain r where r: "r \<in> dyadics \<inter> {0..1}" "r < 1" "\<bar>r-1\<bar> < \<epsilon>/2" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
120 |
by (smt (verit) \<open>0 < \<epsilon>\<close> half_gt_zero) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
121 |
define G' where "G' \<equiv> topspace X - X closure_of G r" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
122 |
show ?thesis |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
123 |
proof (intro exI conjI) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
124 |
show "openin X G'" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
125 |
unfolding G'_def by fastforce |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
126 |
obtain r' where "r' \<in> dyadics \<and> 0 \<le> r' \<and> r' \<le> 1 \<and> r < r' \<and> \<bar>r' - r\<bar> < 1 - r" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
127 |
using B r by force |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
128 |
moreover |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
129 |
have "1 - r \<in> dyadics" "0 \<le> r" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
130 |
using 1 r dyadics_diff by force+ |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
131 |
ultimately have "x \<notin> X closure_of G r" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
132 |
using G True r fle by force |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
133 |
then show "x \<in> G'" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
134 |
by (simp add: G'_def that) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
135 |
show "\<forall>y \<in> G'. \<bar>f y - f x\<bar> < \<epsilon>" |
78202 | 136 |
using ** f_le1 in_closure_of r by (fastforce simp: True G'_def) |
78127
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
137 |
qed |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
138 |
next |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
139 |
case False |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
140 |
have "0 < f x" "f x < 1" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
141 |
using fle1 f_ge that(1) \<open>f x \<noteq> 0\<close> \<open>f x \<noteq> 1\<close> by (metis order_le_less) + |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
142 |
obtain r where r: "r \<in> dyadics \<inter> {0..1}" "r < f x" "\<bar>r - f x\<bar> < \<epsilon> / 2" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
143 |
using A \<open>0 < \<epsilon>\<close> \<open>0 < f x\<close> \<open>f x < 1\<close> by (smt (verit, best) half_gt_zero) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
144 |
obtain r' where r': "r' \<in> dyadics \<inter> {0..1}" "f x < r'" "\<bar>r' - f x\<bar> < \<epsilon> / 2" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
145 |
using B \<open>0 < \<epsilon>\<close> \<open>0 < f x\<close> \<open>f x < 1\<close> by (smt (verit, best) half_gt_zero) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
146 |
have "r < 1" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
147 |
using \<open>f x < 1\<close> r(2) by force |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
148 |
show ?thesis |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
149 |
proof (intro conjI exI) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
150 |
show "openin X (G r' - X closure_of G r)" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
151 |
using closedin_closure_of opeG r' by blast |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
152 |
have "x \<in> X closure_of G r \<Longrightarrow> False" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
153 |
using B [of r "f x - r"] r \<open>r < 1\<close> G [of r] fle by force |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
154 |
then show "x \<in> G r' - X closure_of G r" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
155 |
using ** r' by fastforce |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
156 |
show "\<forall>y\<in>G r' - X closure_of G r. \<bar>f y - f x\<bar> < \<epsilon>" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
157 |
using r r' ** G closure_of_subset field_sum_of_halves fle openin_subset subset_eq |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
158 |
by (smt (verit) DiffE opeG) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
159 |
qed |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
160 |
qed |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
161 |
qed |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
162 |
qed |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
163 |
then have contf: "continuous_map X (top_of_set {0..1}) f" |
78202 | 164 |
by (force simp: Met_TC.continuous_map_to_metric dist_real_def continuous_map_in_subtopology fim simp flip: mtopology_is_euclidean) |
78127
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
165 |
define g where "g \<equiv> \<lambda>x. a + (b - a) * f x" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
166 |
show thesis |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
167 |
proof |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
168 |
have "continuous_map X euclideanreal g" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
169 |
using contf \<open>a \<le> b\<close> unfolding g_def by (auto simp: continuous_intros continuous_map_in_subtopology) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
170 |
moreover have "g ` (topspace X) \<subseteq> {a..b}" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
171 |
using mult_left_le [of "f _" "b-a"] contf \<open>a \<le> b\<close> |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
172 |
by (simp add: g_def add.commute continuous_map_in_subtopology image_subset_iff le_diff_eq) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
173 |
ultimately show "continuous_map X (top_of_set {a..b}) g" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
174 |
by (meson continuous_map_in_subtopology) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
175 |
show "g ` S \<subseteq> {a}" "g ` T \<subseteq> {b}" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
176 |
using fimS fimT by (auto simp: g_def) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
177 |
qed |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
178 |
qed |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
179 |
|
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
180 |
lemma Urysohn_lemma_alt: |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
181 |
fixes a b :: real |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
182 |
assumes "normal_space X" "closedin X S" "closedin X T" "disjnt S T" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
183 |
obtains f where "continuous_map X euclideanreal f" "f ` S \<subseteq> {a}" "f ` T \<subseteq> {b}" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
184 |
by (metis Urysohn_lemma assms continuous_map_in_subtopology disjnt_sym linear) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
185 |
|
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
186 |
lemma normal_space_iff_Urysohn_gen_alt: |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
187 |
assumes "a \<noteq> b" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
188 |
shows "normal_space X \<longleftrightarrow> |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
189 |
(\<forall>S T. closedin X S \<and> closedin X T \<and> disjnt S T |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
190 |
\<longrightarrow> (\<exists>f. continuous_map X euclideanreal f \<and> f ` S \<subseteq> {a} \<and> f ` T \<subseteq> {b}))" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
191 |
(is "?lhs=?rhs") |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
192 |
proof |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
193 |
show "?lhs \<Longrightarrow> ?rhs" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
194 |
by (metis Urysohn_lemma_alt) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
195 |
next |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
196 |
assume R: ?rhs |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
197 |
show ?lhs |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
198 |
unfolding normal_space_def |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
199 |
proof clarify |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
200 |
fix S T |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
201 |
assume "closedin X S" and "closedin X T" and "disjnt S T" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
202 |
with R obtain f where contf: "continuous_map X euclideanreal f" and "f ` S \<subseteq> {a}" "f ` T \<subseteq> {b}" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
203 |
by meson |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
204 |
show "\<exists>U V. openin X U \<and> openin X V \<and> S \<subseteq> U \<and> T \<subseteq> V \<and> disjnt U V" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
205 |
proof (intro conjI exI) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
206 |
show "openin X {x \<in> topspace X. f x \<in> ball a (\<bar>a - b\<bar> / 2)}" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
207 |
by (force intro!: openin_continuous_map_preimage [OF contf]) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
208 |
show "openin X {x \<in> topspace X. f x \<in> ball b (\<bar>a - b\<bar> / 2)}" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
209 |
by (force intro!: openin_continuous_map_preimage [OF contf]) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
210 |
show "S \<subseteq> {x \<in> topspace X. f x \<in> ball a (\<bar>a - b\<bar> / 2)}" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
211 |
using \<open>closedin X S\<close> closedin_subset \<open>f ` S \<subseteq> {a}\<close> assms by force |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
212 |
show "T \<subseteq> {x \<in> topspace X. f x \<in> ball b (\<bar>a - b\<bar> / 2)}" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
213 |
using \<open>closedin X T\<close> closedin_subset \<open>f ` T \<subseteq> {b}\<close> assms by force |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
214 |
have "\<And>x. \<lbrakk>x \<in> topspace X; dist a (f x) < \<bar>a-b\<bar>/2; dist b (f x) < \<bar>a-b\<bar>/2\<rbrakk> \<Longrightarrow> False" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
215 |
by (smt (verit, best) dist_real_def dist_triangle_half_l) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
216 |
then show "disjnt {x \<in> topspace X. f x \<in> ball a (\<bar>a-b\<bar> / 2)} {x \<in> topspace X. f x \<in> ball b (\<bar>a-b\<bar> / 2)}" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
217 |
using disjnt_iff by fastforce |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
218 |
qed |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
219 |
qed |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
220 |
qed |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
221 |
|
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
222 |
lemma normal_space_iff_Urysohn_gen: |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
223 |
fixes a b::real |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
224 |
shows |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
225 |
"a < b \<Longrightarrow> |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
226 |
normal_space X \<longleftrightarrow> |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
227 |
(\<forall>S T. closedin X S \<and> closedin X T \<and> disjnt S T |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
228 |
\<longrightarrow> (\<exists>f. continuous_map X (top_of_set {a..b}) f \<and> |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
229 |
f ` S \<subseteq> {a} \<and> f ` T \<subseteq> {b}))" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
230 |
by (metis linear not_le Urysohn_lemma normal_space_iff_Urysohn_gen_alt continuous_map_in_subtopology) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
231 |
|
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
232 |
lemma normal_space_iff_Urysohn_alt: |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
233 |
"normal_space X \<longleftrightarrow> |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
234 |
(\<forall>S T. closedin X S \<and> closedin X T \<and> disjnt S T |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
235 |
\<longrightarrow> (\<exists>f. continuous_map X euclideanreal f \<and> |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
236 |
f ` S \<subseteq> {0} \<and> f ` T \<subseteq> {1}))" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
237 |
by (rule normal_space_iff_Urysohn_gen_alt) auto |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
238 |
|
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
239 |
lemma normal_space_iff_Urysohn: |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
240 |
"normal_space X \<longleftrightarrow> |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
241 |
(\<forall>S T. closedin X S \<and> closedin X T \<and> disjnt S T |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
242 |
\<longrightarrow> (\<exists>f::'a\<Rightarrow>real. continuous_map X (top_of_set {0..1}) f \<and> |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
243 |
f ` S \<subseteq> {0} \<and> f ` T \<subseteq> {1}))" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
244 |
by (rule normal_space_iff_Urysohn_gen) auto |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
245 |
|
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
246 |
lemma normal_space_perfect_map_image: |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
247 |
"\<lbrakk>normal_space X; perfect_map X Y f\<rbrakk> \<Longrightarrow> normal_space Y" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
248 |
unfolding perfect_map_def proper_map_def |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
249 |
using normal_space_continuous_closed_map_image by fastforce |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
250 |
|
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
251 |
lemma Hausdorff_normal_space_closed_continuous_map_image: |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
252 |
"\<lbrakk>normal_space X; closed_map X Y f; continuous_map X Y f; |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
253 |
f ` topspace X = topspace Y; t1_space Y\<rbrakk> |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
254 |
\<Longrightarrow> Hausdorff_space Y" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
255 |
by (metis normal_space_continuous_closed_map_image normal_t1_imp_Hausdorff_space) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
256 |
|
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
257 |
lemma normal_Hausdorff_space_closed_continuous_map_image: |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
258 |
"\<lbrakk>normal_space X; Hausdorff_space X; closed_map X Y f; |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
259 |
continuous_map X Y f; f ` topspace X = topspace Y\<rbrakk> |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
260 |
\<Longrightarrow> normal_space Y \<and> Hausdorff_space Y" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
261 |
by (meson normal_space_continuous_closed_map_image normal_t1_eq_Hausdorff_space t1_space_closed_map_image) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
262 |
|
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
263 |
lemma Lindelof_cover: |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
264 |
assumes "regular_space X" and "Lindelof_space X" and "S \<noteq> {}" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
265 |
and clo: "closedin X S" "closedin X T" "disjnt S T" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
266 |
obtains h :: "nat \<Rightarrow> 'a set" where |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
267 |
"\<And>n. openin X (h n)" "\<And>n. disjnt T (X closure_of (h n))" and "S \<subseteq> \<Union>(range h)" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
268 |
proof - |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
269 |
have "\<exists>U. openin X U \<and> x \<in> U \<and> disjnt T (X closure_of U)" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
270 |
if "x \<in> S" for x |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
271 |
using \<open>regular_space X\<close> unfolding regular_space |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
272 |
by (metis (full_types) Diff_iff \<open>disjnt S T\<close> clo closure_of_eq disjnt_iff in_closure_of that) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
273 |
then obtain h where oh: "\<And>x. x \<in> S \<Longrightarrow> openin X (h x)" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
274 |
and xh: "\<And>x. x \<in> S \<Longrightarrow> x \<in> h x" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
275 |
and dh: "\<And>x. x \<in> S \<Longrightarrow> disjnt T (X closure_of h x)" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
276 |
by metis |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
277 |
have "Lindelof_space(subtopology X S)" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
278 |
by (simp add: Lindelof_space_closedin_subtopology \<open>Lindelof_space X\<close> \<open>closedin X S\<close>) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
279 |
then obtain \<U> where \<U>: "countable \<U> \<and> \<U> \<subseteq> h ` S \<and> S \<subseteq> \<Union>\<U>" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
280 |
unfolding Lindelof_space_subtopology_subset [OF closedin_subset [OF \<open>closedin X S\<close>]] |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
281 |
by (smt (verit, del_insts) oh xh UN_I image_iff subsetI) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
282 |
with \<open>S \<noteq> {}\<close> have "\<U> \<noteq> {}" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
283 |
by blast |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
284 |
show ?thesis |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
285 |
proof |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
286 |
show "openin X (from_nat_into \<U> n)" for n |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
287 |
by (metis \<U> from_nat_into image_iff \<open>\<U> \<noteq> {}\<close> oh subsetD) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
288 |
show "disjnt T (X closure_of (from_nat_into \<U>) n)" for n |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
289 |
using dh from_nat_into [OF \<open>\<U> \<noteq> {}\<close>] |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
290 |
by (metis \<U> f_inv_into_f inv_into_into subset_eq) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
291 |
show "S \<subseteq> \<Union>(range (from_nat_into \<U>))" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
292 |
by (simp add: \<U> \<open>\<U> \<noteq> {}\<close>) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
293 |
qed |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
294 |
qed |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
295 |
|
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
296 |
lemma regular_Lindelof_imp_normal_space: |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
297 |
assumes "regular_space X" and "Lindelof_space X" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
298 |
shows "normal_space X" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
299 |
unfolding normal_space_def |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
300 |
proof clarify |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
301 |
fix S T |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
302 |
assume clo: "closedin X S" "closedin X T" and "disjnt S T" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
303 |
show "\<exists>U V. openin X U \<and> openin X V \<and> S \<subseteq> U \<and> T \<subseteq> V \<and> disjnt U V" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
304 |
proof (cases "S={} \<or> T={}") |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
305 |
case True |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
306 |
with clo show ?thesis |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
307 |
by (meson closedin_def disjnt_empty1 disjnt_empty2 openin_empty openin_topspace subset_empty) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
308 |
next |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
309 |
case False |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
310 |
obtain h :: "nat \<Rightarrow> 'a set" where |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
311 |
opeh: "\<And>n. openin X (h n)" and dish: "\<And>n. disjnt T (X closure_of (h n))" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
312 |
and Sh: "S \<subseteq> \<Union>(range h)" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
313 |
by (metis Lindelof_cover False \<open>disjnt S T\<close> assms clo) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
314 |
obtain k :: "nat \<Rightarrow> 'a set" where |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
315 |
opek: "\<And>n. openin X (k n)" and disk: "\<And>n. disjnt S (X closure_of (k n))" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
316 |
and Tk: "T \<subseteq> \<Union>(range k)" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
317 |
by (metis Lindelof_cover False \<open>disjnt S T\<close> assms clo disjnt_sym) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
318 |
define U where "U \<equiv> \<Union>i. h i - (\<Union>j<i. X closure_of k j)" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
319 |
define V where "V \<equiv> \<Union>i. k i - (\<Union>j\<le>i. X closure_of h j)" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
320 |
show ?thesis |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
321 |
proof (intro exI conjI) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
322 |
show "openin X U" "openin X V" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
323 |
unfolding U_def V_def |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
324 |
by (force intro!: opek opeh closedin_Union closedin_closure_of)+ |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
325 |
show "S \<subseteq> U" "T \<subseteq> V" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
326 |
using Sh Tk dish disk by (fastforce simp: U_def V_def disjnt_iff)+ |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
327 |
have "\<And>x i j. \<lbrakk>x \<in> k i; x \<in> h j; \<forall>j\<le>i. x \<notin> X closure_of h j\<rbrakk> |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
328 |
\<Longrightarrow> \<exists>i<j. x \<in> X closure_of k i" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
329 |
by (metis in_closure_of linorder_not_less opek openin_subset subsetD) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
330 |
then show "disjnt U V" |
78202 | 331 |
by (force simp: U_def V_def disjnt_iff) |
78127
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
332 |
qed |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
333 |
qed |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
334 |
qed |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
335 |
|
78274
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
336 |
theorem Tietze_extension_closed_real_interval: |
78127
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
337 |
assumes "normal_space X" and "closedin X S" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
338 |
and contf: "continuous_map (subtopology X S) euclideanreal f" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
339 |
and fim: "f ` S \<subseteq> {a..b}" and "a \<le> b" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
340 |
obtains g |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
341 |
where "continuous_map X euclideanreal g" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
342 |
"\<And>x. x \<in> S \<Longrightarrow> g x = f x" "g ` topspace X \<subseteq> {a..b}" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
343 |
proof - |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
344 |
define c where "c \<equiv> max \<bar>a\<bar> \<bar>b\<bar> + 1" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
345 |
have "0 < c" and c: "\<And>x. x \<in> S \<Longrightarrow> \<bar>f x\<bar> \<le> c" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
346 |
using fim by (auto simp: c_def image_subset_iff) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
347 |
define good where |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
348 |
"good \<equiv> \<lambda>g n. continuous_map X euclideanreal g \<and> (\<forall>x \<in> S. \<bar>f x - g x\<bar> \<le> c * (2/3)^n)" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
349 |
have step: "\<exists>g. good g (Suc n) \<and> |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
350 |
(\<forall>x \<in> topspace X. \<bar>g x - h x\<bar> \<le> c * (2/3)^n / 3)" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
351 |
if h: "good h n" for n h |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
352 |
proof - |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
353 |
have pos: "0 < c * (2/3) ^ n" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
354 |
by (simp add: \<open>0 < c\<close>) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
355 |
have S_eq: "S = topspace(subtopology X S)" and "S \<subseteq> topspace X" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
356 |
using \<open>closedin X S\<close> closedin_subset by auto |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
357 |
define d where "d \<equiv> c/3 * (2/3) ^ n" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
358 |
define SA where "SA \<equiv> {x \<in> S. f x - h x \<in> {..-d}}" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
359 |
define SB where "SB \<equiv> {x \<in> S. f x - h x \<in> {d..}}" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
360 |
have contfh: "continuous_map (subtopology X S) euclideanreal (\<lambda>x. f x - h x)" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
361 |
using that |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
362 |
by (simp add: contf good_def continuous_map_diff continuous_map_from_subtopology) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
363 |
then have "closedin (subtopology X S) SA" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
364 |
unfolding SA_def |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
365 |
by (smt (verit, del_insts) closed_closedin continuous_map_closedin Collect_cong S_eq closed_real_atMost) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
366 |
then have "closedin X SA" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
367 |
using \<open>closedin X S\<close> closedin_trans_full by blast |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
368 |
moreover have "closedin (subtopology X S) SB" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
369 |
unfolding SB_def |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
370 |
using closedin_continuous_map_preimage_gen [OF contfh] |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
371 |
by (metis (full_types) S_eq closed_atLeast closed_closedin closedin_topspace) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
372 |
then have "closedin X SB" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
373 |
using \<open>closedin X S\<close> closedin_trans_full by blast |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
374 |
moreover have "disjnt SA SB" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
375 |
using pos by (auto simp: d_def disjnt_def SA_def SB_def) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
376 |
moreover have "-d \<le> d" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
377 |
using pos by (auto simp: d_def) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
378 |
ultimately |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
379 |
obtain g where contg: "continuous_map X (top_of_set {- d..d}) g" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
380 |
and ga: "g ` SA \<subseteq> {- d}" and gb: "g ` SB \<subseteq> {d}" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
381 |
using Urysohn_lemma \<open>normal_space X\<close> by metis |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
382 |
then have g_le_d: "\<And>x. x \<in> topspace X \<Longrightarrow> \<bar>g x\<bar> \<le> d" |
78320
eb9a9690b8f5
cosmetic improvements, new lemmas, especially more uses of function space
paulson <lp15@cam.ac.uk>
parents:
78283
diff
changeset
|
383 |
by (fastforce simp: abs_le_iff continuous_map_def minus_le_iff) |
78127
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
384 |
have g_eq_d: "\<And>x. \<lbrakk>x \<in> S; f x - h x \<le> -d\<rbrakk> \<Longrightarrow> g x = -d" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
385 |
using ga by (auto simp: SA_def) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
386 |
have g_eq_negd: "\<And>x. \<lbrakk>x \<in> S; f x - h x \<ge> d\<rbrakk> \<Longrightarrow> g x = d" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
387 |
using gb by (auto simp: SB_def) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
388 |
show ?thesis |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
389 |
unfolding good_def |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
390 |
proof (intro conjI strip exI) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
391 |
show "continuous_map X euclideanreal (\<lambda>x. h x + g x)" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
392 |
using contg continuous_map_add continuous_map_in_subtopology that |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
393 |
unfolding good_def by blast |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
394 |
show "\<bar>f x - (h x + g x)\<bar> \<le> c * (2 / 3) ^ Suc n" if "x \<in> S" for x |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
395 |
proof - |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
396 |
have x: "x \<in> topspace X" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
397 |
using \<open>S \<subseteq> topspace X\<close> that by auto |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
398 |
have "\<bar>f x - h x\<bar> \<le> c * (2/3) ^ n" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
399 |
using good_def h that by blast |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
400 |
with g_eq_d [OF that] g_eq_negd [OF that] g_le_d [OF x] |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
401 |
have "\<bar>f x - (h x + g x)\<bar> \<le> d + d" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
402 |
unfolding d_def by linarith |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
403 |
then show ?thesis |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
404 |
by (simp add: d_def) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
405 |
qed |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
406 |
show "\<bar>h x + g x - h x\<bar> \<le> c * (2 / 3) ^ n / 3" if "x \<in> topspace X" for x |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
407 |
using that d_def g_le_d by auto |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
408 |
qed |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
409 |
qed |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
410 |
then obtain nxtg where nxtg: "\<And>h n. good h n \<Longrightarrow> |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
411 |
good (nxtg h n) (Suc n) \<and> (\<forall>x \<in> topspace X. \<bar>nxtg h n x - h x\<bar> \<le> c * (2/3)^n / 3)" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
412 |
by metis |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
413 |
define g where "g \<equiv> rec_nat (\<lambda>x. 0) (\<lambda>n r. nxtg r n)" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
414 |
have [simp]: "g 0 x = 0" for x |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
415 |
by (auto simp: g_def) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
416 |
have g_Suc: "g(Suc n) = nxtg (g n) n" for n |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
417 |
by (auto simp: g_def) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
418 |
have good: "good (g n) n" for n |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
419 |
proof (induction n) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
420 |
case 0 |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
421 |
with c show ?case |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
422 |
by (auto simp: good_def) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
423 |
qed (simp add: g_Suc nxtg) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
424 |
have *: "\<And>n x. x \<in> topspace X \<Longrightarrow> \<bar>g(Suc n) x - g n x\<bar> \<le> c * (2/3) ^ n / 3" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
425 |
using nxtg g_Suc good by force |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
426 |
obtain h where conth: "continuous_map X euclideanreal h" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
427 |
and h: "\<And>\<epsilon>. 0 < \<epsilon> \<Longrightarrow> \<forall>\<^sub>F n in sequentially. \<forall>x\<in>topspace X. dist (g n x) (h x) < \<epsilon>" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
428 |
proof (rule Met_TC.continuous_map_uniformly_Cauchy_limit) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
429 |
show "\<forall>\<^sub>F n in sequentially. continuous_map X (Met_TC.mtopology) (g n)" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
430 |
using good good_def by fastforce |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
431 |
show "\<exists>N. \<forall>m n x. N \<le> m \<longrightarrow> N \<le> n \<longrightarrow> x \<in> topspace X \<longrightarrow> dist (g m x) (g n x) < \<epsilon>" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
432 |
if "\<epsilon> > 0" for \<epsilon> |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
433 |
proof - |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
434 |
have "\<forall>\<^sub>F n in sequentially. \<bar>(2/3) ^ n\<bar> < \<epsilon>/c" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
435 |
proof (rule Archimedean_eventually_pow_inverse) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
436 |
show "0 < \<epsilon> / c" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
437 |
by (simp add: \<open>0 < c\<close> that) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
438 |
qed auto |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
439 |
then obtain N where N: "\<And>n. n \<ge> N \<Longrightarrow> \<bar>(2/3) ^ n\<bar> < \<epsilon>/c" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
440 |
by (meson eventually_sequentially order_le_less_trans) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
441 |
have "\<bar>g m x - g n x\<bar> < \<epsilon>" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
442 |
if "N \<le> m" "N \<le> n" and x: "x \<in> topspace X" "m \<le> n" for m n x |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
443 |
proof (cases "m < n") |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
444 |
case True |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
445 |
have 23: "(\<Sum>k = m..<n. (2/3)^k) = 3 * ((2/3) ^ m - (2/3::real) ^ n)" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
446 |
using \<open>m \<le> n\<close> |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
447 |
by (induction n) (auto simp: le_Suc_eq) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
448 |
have "\<bar>g m x - g n x\<bar> \<le> \<bar>\<Sum>k = m..<n. g (Suc k) x - g k x\<bar>" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
449 |
by (subst sum_Suc_diff' [OF \<open>m \<le> n\<close>]) linarith |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
450 |
also have "\<dots> \<le> (\<Sum>k = m..<n. \<bar>g (Suc k) x - g k x\<bar>)" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
451 |
by (rule sum_abs) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
452 |
also have "\<dots> \<le> (\<Sum>k = m..<n. c * (2/3)^k / 3)" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
453 |
by (meson "*" sum_mono x(1)) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
454 |
also have "\<dots> = (c/3) * (\<Sum>k = m..<n. (2/3)^k)" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
455 |
by (simp add: sum_distrib_left) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
456 |
also have "\<dots> = (c/3) * 3 * ((2/3) ^ m - (2/3) ^ n)" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
457 |
by (simp add: sum_distrib_left 23) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
458 |
also have "... < (c/3) * 3 * ((2/3) ^ m)" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
459 |
using \<open>0 < c\<close> by auto |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
460 |
also have "\<dots> < \<epsilon>" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
461 |
using N [OF \<open>N \<le> m\<close>] \<open>0 < c\<close> by (simp add: field_simps) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
462 |
finally show ?thesis . |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
463 |
qed (use \<open>0 < \<epsilon>\<close> \<open>m \<le> n\<close> in auto) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
464 |
then show ?thesis |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
465 |
by (metis dist_commute_lessI dist_real_def nle_le) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
466 |
qed |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
467 |
qed auto |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
468 |
define \<phi> where "\<phi> \<equiv> \<lambda>x. max a (min (h x) b)" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
469 |
show thesis |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
470 |
proof |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
471 |
show "continuous_map X euclidean \<phi>" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
472 |
unfolding \<phi>_def using conth by (intro continuous_intros) auto |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
473 |
show "\<phi> x = f x" if "x \<in> S" for x |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
474 |
proof - |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
475 |
have x: "x \<in> topspace X" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
476 |
using \<open>closedin X S\<close> closedin_subset that by blast |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
477 |
have "h x = f x" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
478 |
proof (rule Met_TC.limitin_metric_unique) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
479 |
show "limitin Met_TC.mtopology (\<lambda>n. g n x) (h x) sequentially" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
480 |
using h x by (force simp: tendsto_iff eventually_sequentially) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
481 |
show "limitin Met_TC.mtopology (\<lambda>n. g n x) (f x) sequentially" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
482 |
proof (clarsimp simp: tendsto_iff) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
483 |
fix \<epsilon>::real |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
484 |
assume "\<epsilon> > 0" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
485 |
then have "\<forall>\<^sub>F n in sequentially. \<bar>(2/3) ^ n\<bar> < \<epsilon>/c" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
486 |
by (intro Archimedean_eventually_pow_inverse) (auto simp: \<open>c > 0\<close>) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
487 |
then show "\<forall>\<^sub>F n in sequentially. dist (g n x) (f x) < \<epsilon>" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
488 |
apply eventually_elim |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
489 |
by (smt (verit) good x good_def \<open>c > 0\<close> dist_real_def mult.commute pos_less_divide_eq that) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
490 |
qed |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
491 |
qed auto |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
492 |
then show ?thesis |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
493 |
using that fim by (auto simp: \<phi>_def) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
494 |
qed |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
495 |
then show "\<phi> ` topspace X \<subseteq> {a..b}" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
496 |
using fim \<open>a \<le> b\<close> by (auto simp: \<phi>_def) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
497 |
qed |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
498 |
qed |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
499 |
|
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
500 |
|
78274
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
501 |
theorem Tietze_extension_realinterval: |
78127
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
502 |
assumes XS: "normal_space X" "closedin X S" and T: "is_interval T" "T \<noteq> {}" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
503 |
and contf: "continuous_map (subtopology X S) euclideanreal f" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
504 |
and "f ` S \<subseteq> T" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
505 |
obtains g where "continuous_map X euclideanreal g" "g ` topspace X \<subseteq> T" "\<And>x. x \<in> S \<Longrightarrow> g x = f x" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
506 |
proof - |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
507 |
define \<Phi> where |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
508 |
"\<Phi> \<equiv> \<lambda>T::real set. \<forall>f. continuous_map (subtopology X S) euclidean f \<longrightarrow> f`S \<subseteq> T |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
509 |
\<longrightarrow> (\<exists>g. continuous_map X euclidean g \<and> g ` topspace X \<subseteq> T \<and> (\<forall>x \<in> S. g x = f x))" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
510 |
have "\<Phi> T" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
511 |
if *: "\<And>T. \<lbrakk>bounded T; is_interval T; T \<noteq> {}\<rbrakk> \<Longrightarrow> \<Phi> T" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
512 |
and "is_interval T" "T \<noteq> {}" for T |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
513 |
unfolding \<Phi>_def |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
514 |
proof (intro strip) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
515 |
fix f |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
516 |
assume contf: "continuous_map (subtopology X S) euclideanreal f" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
517 |
and "f ` S \<subseteq> T" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
518 |
have \<Phi>T: "\<Phi> ((\<lambda>x. x / (1 + \<bar>x\<bar>)) ` T)" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
519 |
proof (rule *) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
520 |
show "bounded ((\<lambda>x. x / (1 + \<bar>x\<bar>)) ` T)" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
521 |
using shrink_range [of T] by (force intro: boundedI [where B=1]) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
522 |
show "is_interval ((\<lambda>x. x / (1 + \<bar>x\<bar>)) ` T)" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
523 |
using connected_shrink that(2) is_interval_connected_1 by blast |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
524 |
show "(\<lambda>x. x / (1 + \<bar>x\<bar>)) ` T \<noteq> {}" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
525 |
using \<open>T \<noteq> {}\<close> by auto |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
526 |
qed |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
527 |
moreover have "continuous_map (subtopology X S) euclidean ((\<lambda>x. x / (1 + \<bar>x\<bar>)) \<circ> f)" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
528 |
by (metis contf continuous_map_compose continuous_map_into_fulltopology continuous_map_real_shrink) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
529 |
moreover have "((\<lambda>x. x / (1 + \<bar>x\<bar>)) \<circ> f) ` S \<subseteq> (\<lambda>x. x / (1 + \<bar>x\<bar>)) ` T" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
530 |
using \<open>f ` S \<subseteq> T\<close> by auto |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
531 |
ultimately obtain g |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
532 |
where contg: "continuous_map X euclidean g" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
533 |
and gim: "g ` topspace X \<subseteq> (\<lambda>x. x / (1 + \<bar>x\<bar>)) ` T" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
534 |
and geq: "\<And>x. x \<in> S \<Longrightarrow> g x = ((\<lambda>x. x / (1 + \<bar>x\<bar>)) \<circ> f) x" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
535 |
using \<Phi>T unfolding \<Phi>_def by force |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
536 |
show "\<exists>g. continuous_map X euclideanreal g \<and> g ` topspace X \<subseteq> T \<and> (\<forall>x\<in>S. g x = f x)" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
537 |
proof (intro conjI exI) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
538 |
have "continuous_map X (top_of_set {-1<..<1}) g" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
539 |
using contg continuous_map_in_subtopology gim shrink_range by blast |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
540 |
then show "continuous_map X euclideanreal ((\<lambda>x. x / (1 - \<bar>x\<bar>)) \<circ> g)" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
541 |
by (rule continuous_map_compose) (auto simp: continuous_on_real_grow) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
542 |
show "((\<lambda>x. x / (1 - \<bar>x\<bar>)) \<circ> g) ` topspace X \<subseteq> T" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
543 |
using gim real_grow_shrink by fastforce |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
544 |
show "\<forall>x\<in>S. ((\<lambda>x. x / (1 - \<bar>x\<bar>)) \<circ> g) x = f x" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
545 |
using geq real_grow_shrink by force |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
546 |
qed |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
547 |
qed |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
548 |
moreover have "\<Phi> T" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
549 |
if "bounded T" "is_interval T" "T \<noteq> {}" for T |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
550 |
unfolding \<Phi>_def |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
551 |
proof (intro strip) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
552 |
fix f |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
553 |
assume contf: "continuous_map (subtopology X S) euclideanreal f" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
554 |
and "f ` S \<subseteq> T" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
555 |
obtain a b where ab: "closure T = {a..b}" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
556 |
by (meson \<open>bounded T\<close> \<open>is_interval T\<close> compact_closure connected_compact_interval_1 |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
557 |
connected_imp_connected_closure is_interval_connected) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
558 |
with \<open>T \<noteq> {}\<close> have "a \<le> b" by auto |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
559 |
have "f ` S \<subseteq> {a..b}" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
560 |
using \<open>f ` S \<subseteq> T\<close> ab closure_subset by auto |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
561 |
then obtain g where contg: "continuous_map X euclideanreal g" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
562 |
and gf: "\<And>x. x \<in> S \<Longrightarrow> g x = f x" and gim: "g ` topspace X \<subseteq> {a..b}" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
563 |
using Tietze_extension_closed_real_interval [OF XS contf _ \<open>a \<le> b\<close>] by metis |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
564 |
define W where "W \<equiv> {x \<in> topspace X. g x \<in> closure T - T}" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
565 |
have "{a..b} - {a, b} \<subseteq> T" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
566 |
using that |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
567 |
by (metis ab atLeastAtMost_diff_ends convex_interior_closure interior_atLeastAtMost_real |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
568 |
interior_subset is_interval_convex) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
569 |
with finite_imp_compact have "compact (closure T - T)" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
570 |
by (metis Diff_eq_empty_iff Diff_insert2 ab finite.emptyI finite_Diff_insert) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
571 |
then have "closedin X W" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
572 |
unfolding W_def using closedin_continuous_map_preimage [OF contg] compact_imp_closed by force |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
573 |
moreover have "disjnt W S" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
574 |
unfolding W_def disjnt_iff using \<open>f ` S \<subseteq> T\<close> gf by blast |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
575 |
ultimately obtain h :: "'a \<Rightarrow> real" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
576 |
where conth: "continuous_map X (top_of_set {0..1}) h" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
577 |
and him: "h ` W \<subseteq> {0}" "h ` S \<subseteq> {1}" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
578 |
by (metis XS normal_space_iff_Urysohn) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
579 |
then have him01: "h ` topspace X \<subseteq> {0..1}" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
580 |
by (meson continuous_map_in_subtopology) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
581 |
obtain z where "z \<in> T" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
582 |
using \<open>T \<noteq> {}\<close> by blast |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
583 |
define g' where "g' \<equiv> \<lambda>x. z + h x * (g x - z)" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
584 |
show "\<exists>g. continuous_map X euclidean g \<and> g ` topspace X \<subseteq> T \<and> (\<forall>x\<in>S. g x = f x)" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
585 |
proof (intro exI conjI) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
586 |
show "continuous_map X euclideanreal g'" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
587 |
unfolding g'_def using contg conth continuous_map_in_subtopology |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
588 |
by (intro continuous_intros) auto |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
589 |
show "g' ` topspace X \<subseteq> T" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
590 |
unfolding g'_def |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
591 |
proof clarify |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
592 |
fix x |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
593 |
assume "x \<in> topspace X" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
594 |
show "z + h x * (g x - z) \<in> T" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
595 |
proof (cases "g x \<in> T") |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
596 |
case True |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
597 |
define w where "w \<equiv> z + h x * (g x - z)" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
598 |
have "\<bar>h x\<bar> * \<bar>g x - z\<bar> \<le> \<bar>g x - z\<bar>" "\<bar>1 - h x\<bar> * \<bar>g x - z\<bar> \<le> \<bar>g x - z\<bar>" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
599 |
using him01 \<open>x \<in> topspace X\<close> by (force simp: intro: mult_left_le_one_le)+ |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
600 |
then consider "z \<le> w \<and> w \<le> g x" | "g x \<le> w \<and> w \<le> z" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
601 |
unfolding w_def by (smt (verit) left_diff_distrib mult_cancel_right2 mult_minus_right zero_less_mult_iff) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
602 |
then show ?thesis |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
603 |
using \<open>is_interval T\<close> unfolding w_def is_interval_1 by (metis True \<open>z \<in> T\<close>) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
604 |
next |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
605 |
case False |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
606 |
then have "g x \<in> closure T" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
607 |
using \<open>x \<in> topspace X\<close> ab gim by blast |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
608 |
then have "h x = 0" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
609 |
using him False \<open>x \<in> topspace X\<close> by (auto simp: W_def image_subset_iff) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
610 |
then show ?thesis |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
611 |
by (simp add: \<open>z \<in> T\<close>) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
612 |
qed |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
613 |
qed |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
614 |
show "\<forall>x\<in>S. g' x = f x" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
615 |
using gf him by (auto simp: W_def g'_def) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
616 |
qed |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
617 |
qed |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
618 |
ultimately show thesis |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
619 |
using assms that unfolding \<Phi>_def by best |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
620 |
qed |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
621 |
|
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
622 |
lemma normal_space_iff_Tietze: |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
623 |
"normal_space X \<longleftrightarrow> |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
624 |
(\<forall>f S. closedin X S \<and> |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
625 |
continuous_map (subtopology X S) euclidean f |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
626 |
\<longrightarrow> (\<exists>g:: 'a \<Rightarrow> real. continuous_map X euclidean g \<and> (\<forall>x \<in> S. g x = f x)))" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
627 |
(is "?lhs \<longleftrightarrow> ?rhs") |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
628 |
proof |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
629 |
assume ?lhs |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
630 |
then show ?rhs |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
631 |
by (metis Tietze_extension_realinterval empty_not_UNIV is_interval_univ subset_UNIV) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
632 |
next |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
633 |
assume R: ?rhs |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
634 |
show ?lhs |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
635 |
unfolding normal_space_iff_Urysohn_alt |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
636 |
proof clarify |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
637 |
fix S T |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
638 |
assume "closedin X S" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
639 |
and "closedin X T" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
640 |
and "disjnt S T" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
641 |
then have cloST: "closedin X (S \<union> T)" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
642 |
by (simp add: closedin_Un) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
643 |
moreover |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
644 |
have "continuous_map (subtopology X (S \<union> T)) euclideanreal (\<lambda>x. if x \<in> S then 0 else 1)" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
645 |
unfolding continuous_map_closedin |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
646 |
proof (intro conjI strip) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
647 |
fix C :: "real set" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
648 |
define D where "D \<equiv> {x \<in> topspace X. if x \<in> S then 0 \<in> C else x \<in> T \<and> 1 \<in> C}" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
649 |
have "D \<in> {{}, S, T, S \<union> T}" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
650 |
unfolding D_def |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
651 |
using closedin_subset [OF \<open>closedin X S\<close>] closedin_subset [OF \<open>closedin X T\<close>] \<open>disjnt S T\<close> |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
652 |
by (auto simp: disjnt_iff) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
653 |
then have "closedin X D" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
654 |
using \<open>closedin X S\<close> \<open>closedin X T\<close> closedin_empty by blast |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
655 |
with closedin_subset_topspace |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
656 |
show "closedin (subtopology X (S \<union> T)) {x \<in> topspace (subtopology X (S \<union> T)). (if x \<in> S then 0::real else 1) \<in> C}" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
657 |
apply (simp add: D_def) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
658 |
by (smt (verit, best) Collect_cong Collect_mono_iff Un_def closedin_subset_topspace) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
659 |
qed auto |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
660 |
ultimately obtain g :: "'a \<Rightarrow> real" where |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
661 |
contg: "continuous_map X euclidean g" and gf: "\<forall>x \<in> S \<union> T. g x = (if x \<in> S then 0 else 1)" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
662 |
using R by blast |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
663 |
then show "\<exists>f. continuous_map X euclideanreal f \<and> f ` S \<subseteq> {0} \<and> f ` T \<subseteq> {1}" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
664 |
by (smt (verit) Un_iff \<open>disjnt S T\<close> disjnt_iff image_subset_iff insert_iff) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
665 |
qed |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
666 |
qed |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
667 |
|
78274
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
668 |
subsection \<open>Random metric space stuff\<close> |
78127
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
669 |
|
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
670 |
lemma metrizable_imp_k_space: |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
671 |
assumes "metrizable_space X" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
672 |
shows "k_space X" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
673 |
proof - |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
674 |
obtain M d where "Metric_space M d" and Xeq: "X = Metric_space.mtopology M d" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
675 |
using assms unfolding metrizable_space_def by metis |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
676 |
then interpret Metric_space M d |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
677 |
by blast |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
678 |
show ?thesis |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
679 |
unfolding k_space Xeq |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
680 |
proof clarsimp |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
681 |
fix S |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
682 |
assume "S \<subseteq> M" and S: "\<forall>K. compactin mtopology K \<longrightarrow> closedin (subtopology mtopology K) (K \<inter> S)" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
683 |
have "l \<in> S" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
684 |
if \<sigma>: "range \<sigma> \<subseteq> S" and l: "limitin mtopology \<sigma> l sequentially" for \<sigma> l |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
685 |
proof - |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
686 |
define K where "K \<equiv> insert l (range \<sigma>)" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
687 |
interpret Submetric M d K |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
688 |
proof |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
689 |
show "K \<subseteq> M" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
690 |
unfolding K_def using l limitin_mspace \<open>S \<subseteq> M\<close> \<sigma> by blast |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
691 |
qed |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
692 |
have "compactin mtopology K" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
693 |
unfolding K_def using \<open>S \<subseteq> M\<close> \<sigma> |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
694 |
by (force intro: compactin_sequence_with_limit [OF l]) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
695 |
then have *: "closedin sub.mtopology (K \<inter> S)" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
696 |
by (simp add: S mtopology_submetric) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
697 |
have "\<sigma> n \<in> K \<inter> S" for n |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
698 |
by (simp add: K_def range_subsetD \<sigma>) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
699 |
moreover have "limitin sub.mtopology \<sigma> l sequentially" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
700 |
using l |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
701 |
unfolding sub.limit_metric_sequentially limit_metric_sequentially |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
702 |
by (force simp: K_def) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
703 |
ultimately have "l \<in> K \<inter> S" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
704 |
by (meson * \<sigma> image_subsetI sub.metric_closedin_iff_sequentially_closed) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
705 |
then show ?thesis |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
706 |
by simp |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
707 |
qed |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
708 |
then show "closedin mtopology S" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
709 |
unfolding metric_closedin_iff_sequentially_closed |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
710 |
using \<open>S \<subseteq> M\<close> by blast |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
711 |
qed |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
712 |
qed |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
713 |
|
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
714 |
lemma (in Metric_space) k_space_mtopology: "k_space mtopology" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
715 |
by (simp add: metrizable_imp_k_space metrizable_space_mtopology) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
716 |
|
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
717 |
lemma k_space_euclideanreal: "k_space (euclidean :: 'a::metric_space topology)" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
718 |
using metrizable_imp_k_space metrizable_space_euclidean by auto |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
719 |
|
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
720 |
|
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
721 |
subsection\<open>Hereditarily normal spaces\<close> |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
722 |
|
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
723 |
lemma hereditarily_B: |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
724 |
assumes "\<And>S T. separatedin X S T |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
725 |
\<Longrightarrow> \<exists>U V. openin X U \<and> openin X V \<and> S \<subseteq> U \<and> T \<subseteq> V \<and> disjnt U V" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
726 |
shows "hereditarily normal_space X" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
727 |
unfolding hereditarily_def |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
728 |
proof (intro strip) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
729 |
fix W |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
730 |
assume "W \<subseteq> topspace X" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
731 |
show "normal_space (subtopology X W)" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
732 |
unfolding normal_space_def |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
733 |
proof clarify |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
734 |
fix S T |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
735 |
assume clo: "closedin (subtopology X W) S" "closedin (subtopology X W) T" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
736 |
and "disjnt S T" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
737 |
then have "separatedin (subtopology X W) S T" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
738 |
by (simp add: separatedin_closed_sets) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
739 |
then obtain U V where "openin X U \<and> openin X V \<and> S \<subseteq> U \<and> T \<subseteq> V \<and> disjnt U V" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
740 |
using assms [of S T] by (meson separatedin_subtopology) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
741 |
then show "\<exists>U V. openin (subtopology X W) U \<and> openin (subtopology X W) V \<and> S \<subseteq> U \<and> T \<subseteq> V \<and> disjnt U V" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
742 |
apply (simp add: openin_subtopology_alt) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
743 |
by (meson clo closedin_imp_subset disjnt_subset1 disjnt_subset2 inf_le2) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
744 |
qed |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
745 |
qed |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
746 |
|
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
747 |
lemma hereditarily_C: |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
748 |
assumes "separatedin X S T" and norm: "\<And>U. openin X U \<Longrightarrow> normal_space (subtopology X U)" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
749 |
shows "\<exists>U V. openin X U \<and> openin X V \<and> S \<subseteq> U \<and> T \<subseteq> V \<and> disjnt U V" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
750 |
proof - |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
751 |
define ST where "ST \<equiv> X closure_of S \<inter> X closure_of T" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
752 |
have subX: "S \<subseteq> topspace X" "T \<subseteq> topspace X" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
753 |
by (meson \<open>separatedin X S T\<close> separation_closedin_Un_gen)+ |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
754 |
have sub: "S \<subseteq> topspace X - ST" "T \<subseteq> topspace X - ST" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
755 |
unfolding ST_def |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
756 |
by (metis Diff_mono Diff_triv \<open>separatedin X S T\<close> Int_lower1 Int_lower2 separatedin_def)+ |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
757 |
have "normal_space (subtopology X (topspace X - ST))" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
758 |
by (simp add: ST_def norm closedin_Int openin_diff) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
759 |
moreover have " disjnt (subtopology X (topspace X - ST) closure_of S) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
760 |
(subtopology X (topspace X - ST) closure_of T)" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
761 |
using Int_absorb1 ST_def sub by (fastforce simp: disjnt_iff closure_of_subtopology) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
762 |
ultimately show ?thesis |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
763 |
using sub subX |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
764 |
apply (simp add: normal_space_closures) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
765 |
by (metis ST_def closedin_Int closedin_closure_of closedin_def openin_trans_full) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
766 |
qed |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
767 |
|
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
768 |
lemma hereditarily_normal_space: |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
769 |
"hereditarily normal_space X \<longleftrightarrow> (\<forall>U. openin X U \<longrightarrow> normal_space(subtopology X U))" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
770 |
by (metis hereditarily_B hereditarily_C hereditarily) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
771 |
|
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
772 |
lemma hereditarily_normal_separation: |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
773 |
"hereditarily normal_space X \<longleftrightarrow> |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
774 |
(\<forall>S T. separatedin X S T |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
775 |
\<longrightarrow> (\<exists>U V. openin X U \<and> openin X V \<and> S \<subseteq> U \<and> T \<subseteq> V \<and> disjnt U V))" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
776 |
by (metis hereditarily_B hereditarily_C hereditarily) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
777 |
|
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
778 |
|
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
779 |
lemma metrizable_imp_hereditarily_normal_space: |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
780 |
"metrizable_space X \<Longrightarrow> hereditarily normal_space X" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
781 |
by (simp add: hereditarily metrizable_imp_normal_space metrizable_space_subtopology) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
782 |
|
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
783 |
lemma metrizable_space_separation: |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
784 |
"\<lbrakk>metrizable_space X; separatedin X S T\<rbrakk> |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
785 |
\<Longrightarrow> \<exists>U V. openin X U \<and> openin X V \<and> S \<subseteq> U \<and> T \<subseteq> V \<and> disjnt U V" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
786 |
by (metis hereditarily hereditarily_C metrizable_imp_hereditarily_normal_space) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
787 |
|
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
788 |
lemma hereditarily_normal_separation_pairwise: |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
789 |
"hereditarily normal_space X \<longleftrightarrow> |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
790 |
(\<forall>\<U>. finite \<U> \<and> (\<forall>S \<in> \<U>. S \<subseteq> topspace X) \<and> pairwise (separatedin X) \<U> |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
791 |
\<longrightarrow> (\<exists>\<F>. (\<forall>S \<in> \<U>. openin X (\<F> S) \<and> S \<subseteq> \<F> S) \<and> |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
792 |
pairwise (\<lambda>S T. disjnt (\<F> S) (\<F> T)) \<U>))" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
793 |
(is "?lhs \<longleftrightarrow> ?rhs") |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
794 |
proof |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
795 |
assume L: ?lhs |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
796 |
show ?rhs |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
797 |
proof clarify |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
798 |
fix \<U> |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
799 |
assume "finite \<U>" and \<U>: "\<forall>S\<in>\<U>. S \<subseteq> topspace X" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
800 |
and pw\<U>: "pairwise (separatedin X) \<U>" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
801 |
have "\<exists>V W. openin X V \<and> openin X W \<and> S \<subseteq> V \<and> |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
802 |
(\<forall>T. T \<in> \<U> \<and> T \<noteq> S \<longrightarrow> T \<subseteq> W) \<and> disjnt V W" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
803 |
if "S \<in> \<U>" for S |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
804 |
proof - |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
805 |
have "separatedin X S (\<Union>(\<U> - {S}))" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
806 |
by (metis \<U> \<open>finite \<U>\<close> pw\<U> finite_Diff pairwise_alt separatedin_Union(1) that) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
807 |
with L show ?thesis |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
808 |
unfolding hereditarily_normal_separation |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
809 |
by (smt (verit) Diff_iff UnionI empty_iff insert_iff subset_iff) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
810 |
qed |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
811 |
then obtain \<F> \<G> |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
812 |
where *: "\<And>S. S \<in> \<U> \<Longrightarrow> S \<subseteq> \<F> S \<and> (\<forall>T. T \<in> \<U> \<and> T \<noteq> S \<longrightarrow> T \<subseteq> \<G> S)" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
813 |
and ope: "\<And>S. S \<in> \<U> \<Longrightarrow> openin X (\<F> S) \<and> openin X (\<G> S)" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
814 |
and dis: "\<And>S. S \<in> \<U> \<Longrightarrow> disjnt (\<F> S) (\<G> S)" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
815 |
by metis |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
816 |
define \<H> where "\<H> \<equiv> \<lambda>S. \<F> S \<inter> (\<Inter>T \<in> \<U> - {S}. \<G> T)" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
817 |
show "\<exists>\<F>. (\<forall>S\<in>\<U>. openin X (\<F> S) \<and> S \<subseteq> \<F> S) \<and> pairwise (\<lambda>S T. disjnt (\<F> S) (\<F> T)) \<U>" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
818 |
proof (intro exI conjI strip) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
819 |
show "openin X (\<H> S)" if "S \<in> \<U>" for S |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
820 |
unfolding \<H>_def |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
821 |
by (smt (verit) ope that DiffD1 \<open>finite \<U>\<close> finite_Diff finite_imageI imageE openin_Int_Inter) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
822 |
show "S \<subseteq> \<H> S" if "S \<in> \<U>" for S |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
823 |
unfolding \<H>_def using "*" that by auto |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
824 |
show "pairwise (\<lambda>S T. disjnt (\<H> S) (\<H> T)) \<U>" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
825 |
using dis by (fastforce simp: disjnt_iff pairwise_alt \<H>_def) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
826 |
qed |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
827 |
qed |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
828 |
next |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
829 |
assume R: ?rhs |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
830 |
show ?lhs |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
831 |
unfolding hereditarily_normal_separation |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
832 |
proof (intro strip) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
833 |
fix S T |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
834 |
assume "separatedin X S T" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
835 |
show "\<exists>U V. openin X U \<and> openin X V \<and> S \<subseteq> U \<and> T \<subseteq> V \<and> disjnt U V" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
836 |
proof (cases "T=S") |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
837 |
case True |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
838 |
then show ?thesis |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
839 |
using \<open>separatedin X S T\<close> by force |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
840 |
next |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
841 |
case False |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
842 |
have "pairwise (separatedin X) {S, T}" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
843 |
by (simp add: \<open>separatedin X S T\<close> pairwise_insert separatedin_sym) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
844 |
moreover have "\<forall>S\<in>{S, T}. S \<subseteq> topspace X" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
845 |
by (metis \<open>separatedin X S T\<close> insertE separatedin_def singletonD) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
846 |
ultimately show ?thesis |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
847 |
using R by (smt (verit) False finite.emptyI finite.insertI insertCI pairwiseD) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
848 |
qed |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
849 |
qed |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
850 |
qed |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
851 |
|
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
852 |
lemma hereditarily_normal_space_perfect_map_image: |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
853 |
"\<lbrakk>hereditarily normal_space X; perfect_map X Y f\<rbrakk> \<Longrightarrow> hereditarily normal_space Y" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
854 |
unfolding perfect_map_def proper_map_def |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
855 |
by (meson hereditarily_normal_space_continuous_closed_map_image) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
856 |
|
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
857 |
lemma regular_second_countable_imp_hereditarily_normal_space: |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
858 |
assumes "regular_space X \<and> second_countable X" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
859 |
shows "hereditarily normal_space X" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
860 |
unfolding hereditarily |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
861 |
proof (intro regular_Lindelof_imp_normal_space strip) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
862 |
show "regular_space (subtopology X S)" for S |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
863 |
by (simp add: assms regular_space_subtopology) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
864 |
show "Lindelof_space (subtopology X S)" for S |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
865 |
using assms by (simp add: second_countable_imp_Lindelof_space second_countable_subtopology) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
866 |
qed |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
867 |
|
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
868 |
|
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
869 |
subsection\<open>Completely regular spaces\<close> |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
870 |
|
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
871 |
definition completely_regular_space where |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
872 |
"completely_regular_space X \<equiv> |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
873 |
\<forall>S x. closedin X S \<and> x \<in> topspace X - S |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
874 |
\<longrightarrow> (\<exists>f::'a\<Rightarrow>real. continuous_map X (top_of_set {0..1}) f \<and> |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
875 |
f x = 0 \<and> (f ` S \<subseteq> {1}))" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
876 |
|
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
877 |
lemma homeomorphic_completely_regular_space_aux: |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
878 |
assumes X: "completely_regular_space X" and hom: "X homeomorphic_space Y" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
879 |
shows "completely_regular_space Y" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
880 |
proof - |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
881 |
obtain f g where hmf: "homeomorphic_map X Y f" and hmg: "homeomorphic_map Y X g" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
882 |
and fg: "(\<forall>x \<in> topspace X. g(f x) = x) \<and> (\<forall>y \<in> topspace Y. f(g y) = y)" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
883 |
using assms X homeomorphic_maps_map homeomorphic_space_def by fastforce |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
884 |
show ?thesis |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
885 |
unfolding completely_regular_space_def |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
886 |
proof clarify |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
887 |
fix S x |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
888 |
assume A: "closedin Y S" and x: "x \<in> topspace Y" and "x \<notin> S" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
889 |
then have "closedin X (g`S)" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
890 |
using hmg homeomorphic_map_closedness_eq by blast |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
891 |
moreover have "g x \<notin> g`S" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
892 |
by (meson A x \<open>x \<notin> S\<close> closedin_subset hmg homeomorphic_imp_injective_map inj_on_image_mem_iff) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
893 |
ultimately obtain \<phi> where \<phi>: "continuous_map X (top_of_set {0..1::real}) \<phi> \<and> \<phi> (g x) = 0 \<and> \<phi> ` g`S \<subseteq> {1}" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
894 |
by (metis DiffI X completely_regular_space_def hmg homeomorphic_imp_surjective_map image_eqI x) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
895 |
then have "continuous_map Y (top_of_set {0..1::real}) (\<phi> \<circ> g)" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
896 |
by (meson continuous_map_compose hmg homeomorphic_imp_continuous_map) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
897 |
then show "\<exists>\<psi>. continuous_map Y (top_of_set {0..1::real}) \<psi> \<and> \<psi> x = 0 \<and> \<psi> ` S \<subseteq> {1}" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
898 |
by (metis \<phi> comp_apply image_comp) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
899 |
qed |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
900 |
qed |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
901 |
|
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
902 |
lemma homeomorphic_completely_regular_space: |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
903 |
assumes "X homeomorphic_space Y" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
904 |
shows "completely_regular_space X \<longleftrightarrow> completely_regular_space Y" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
905 |
by (meson assms homeomorphic_completely_regular_space_aux homeomorphic_space_sym) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
906 |
|
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
907 |
lemma completely_regular_space_alt: |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
908 |
"completely_regular_space X \<longleftrightarrow> |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
909 |
(\<forall>S x. closedin X S \<longrightarrow> x \<in> topspace X - S |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
910 |
\<longrightarrow> (\<exists>f. continuous_map X euclideanreal f \<and> f x = 0 \<and> f ` S \<subseteq> {1}))" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
911 |
proof - |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
912 |
have "\<exists>f. continuous_map X (top_of_set {0..1::real}) f \<and> f x = 0 \<and> f ` S \<subseteq> {1}" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
913 |
if "closedin X S" "x \<in> topspace X - S" and f: "continuous_map X euclideanreal f \<and> f x = 0 \<and> f ` S \<subseteq> {1}" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
914 |
for S x f |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
915 |
proof (intro exI conjI) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
916 |
show "continuous_map X (top_of_set {0..1}) (\<lambda>x. max 0 (min (f x) 1))" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
917 |
using that |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
918 |
by (auto simp: continuous_map_in_subtopology intro!: continuous_map_real_max continuous_map_real_min) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
919 |
qed (use that in auto) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
920 |
with continuous_map_in_subtopology show ?thesis |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
921 |
unfolding completely_regular_space_def by metis |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
922 |
qed |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
923 |
|
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
924 |
text \<open>As above, but with @{term openin}\<close> |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
925 |
lemma completely_regular_space_alt': |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
926 |
"completely_regular_space X \<longleftrightarrow> |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
927 |
(\<forall>S x. openin X S \<longrightarrow> x \<in> S |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
928 |
\<longrightarrow> (\<exists>f. continuous_map X euclideanreal f \<and> f x = 0 \<and> f ` (topspace X - S) \<subseteq> {1}))" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
929 |
apply (simp add: completely_regular_space_alt all_closedin) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
930 |
by (meson openin_subset subsetD) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
931 |
|
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
932 |
lemma completely_regular_space_gen_alt: |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
933 |
fixes a b::real |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
934 |
assumes "a \<noteq> b" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
935 |
shows "completely_regular_space X \<longleftrightarrow> |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
936 |
(\<forall>S x. closedin X S \<longrightarrow> x \<in> topspace X - S |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
937 |
\<longrightarrow> (\<exists>f. continuous_map X euclidean f \<and> f x = a \<and> (f ` S \<subseteq> {b})))" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
938 |
proof - |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
939 |
have "\<exists>f. continuous_map X euclideanreal f \<and> f x = 0 \<and> f ` S \<subseteq> {1}" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
940 |
if "closedin X S" "x \<in> topspace X - S" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
941 |
and f: "continuous_map X euclidean f \<and> f x = a \<and> f ` S \<subseteq> {b}" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
942 |
for S x f |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
943 |
proof (intro exI conjI) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
944 |
show "continuous_map X euclideanreal ((\<lambda>x. inverse(b - a) * (x - a)) \<circ> f)" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
945 |
using that by (intro continuous_intros) auto |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
946 |
qed (use that assms in auto) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
947 |
moreover |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
948 |
have "\<exists>f. continuous_map X euclidean f \<and> f x = a \<and> f ` S \<subseteq> {b}" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
949 |
if "closedin X S" "x \<in> topspace X - S" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
950 |
and f: "continuous_map X euclideanreal f \<and> f x = 0 \<and> f ` S \<subseteq> {1}" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
951 |
for S x f |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
952 |
proof (intro exI conjI) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
953 |
show "continuous_map X euclideanreal ((\<lambda>x. a + (b - a) * x) \<circ> f)" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
954 |
using that by (intro continuous_intros) auto |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
955 |
qed (use that in auto) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
956 |
ultimately show ?thesis |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
957 |
unfolding completely_regular_space_alt by meson |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
958 |
qed |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
959 |
|
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
960 |
text \<open>As above, but with @{term openin}\<close> |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
961 |
lemma completely_regular_space_gen_alt': |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
962 |
fixes a b::real |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
963 |
assumes "a \<noteq> b" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
964 |
shows "completely_regular_space X \<longleftrightarrow> |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
965 |
(\<forall>S x. openin X S \<longrightarrow> x \<in> S |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
966 |
\<longrightarrow> (\<exists>f. continuous_map X euclidean f \<and> f x = a \<and> (f ` (topspace X - S) \<subseteq> {b})))" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
967 |
apply (simp add: completely_regular_space_gen_alt[OF assms] all_closedin) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
968 |
by (meson openin_subset subsetD) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
969 |
|
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
970 |
lemma completely_regular_space_gen: |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
971 |
fixes a b::real |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
972 |
assumes "a < b" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
973 |
shows "completely_regular_space X \<longleftrightarrow> |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
974 |
(\<forall>S x. closedin X S \<and> x \<in> topspace X - S |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
975 |
\<longrightarrow> (\<exists>f. continuous_map X (top_of_set {a..b}) f \<and> f x = a \<and> f ` S \<subseteq> {b}))" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
976 |
proof - |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
977 |
have "\<exists>f. continuous_map X (top_of_set {a..b}) f \<and> f x = a \<and> f ` S \<subseteq> {b}" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
978 |
if "closedin X S" "x \<in> topspace X - S" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
979 |
and f: "continuous_map X euclidean f \<and> f x = a \<and> f ` S \<subseteq> {b}" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
980 |
for S x f |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
981 |
proof (intro exI conjI) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
982 |
show "continuous_map X (top_of_set {a..b}) (\<lambda>x. max a (min (f x) b))" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
983 |
using that assms |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
984 |
by (auto simp: continuous_map_in_subtopology intro!: continuous_map_real_max continuous_map_real_min) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
985 |
qed (use that assms in auto) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
986 |
with continuous_map_in_subtopology assms show ?thesis |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
987 |
using completely_regular_space_gen_alt [of a b] |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
988 |
by (smt (verit) atLeastAtMost_singleton atLeastatMost_empty singletonI) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
989 |
qed |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
990 |
|
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
991 |
lemma normal_imp_completely_regular_space_A: |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
992 |
assumes "normal_space X" "t1_space X" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
993 |
shows "completely_regular_space X" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
994 |
unfolding completely_regular_space_alt |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
995 |
proof clarify |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
996 |
fix x S |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
997 |
assume A: "closedin X S" "x \<notin> S" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
998 |
assume "x \<in> topspace X" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
999 |
then have "closedin X {x}" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1000 |
by (simp add: \<open>t1_space X\<close> closedin_t1_singleton) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1001 |
with A \<open>normal_space X\<close> have "\<exists>f. continuous_map X euclideanreal f \<and> f ` {x} \<subseteq> {0} \<and> f ` S \<subseteq> {1}" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1002 |
using assms unfolding normal_space_iff_Urysohn_alt disjnt_iff by blast |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1003 |
then show "\<exists>f. continuous_map X euclideanreal f \<and> f x = 0 \<and> f ` S \<subseteq> {1}" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1004 |
by auto |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1005 |
qed |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1006 |
|
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1007 |
lemma normal_imp_completely_regular_space_B: |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1008 |
assumes "normal_space X" "regular_space X" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1009 |
shows "completely_regular_space X" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1010 |
unfolding completely_regular_space_alt |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1011 |
proof clarify |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1012 |
fix x S |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1013 |
assume "closedin X S" "x \<notin> S" "x \<in> topspace X" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1014 |
then obtain U C where "openin X U" "closedin X C" "x \<in> U" "U \<subseteq> C" "C \<subseteq> topspace X - S" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1015 |
using assms |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1016 |
unfolding neighbourhood_base_of_closedin [symmetric] neighbourhood_base_of closedin_def by (metis Diff_iff) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1017 |
then obtain f where "continuous_map X euclideanreal f \<and> f ` C \<subseteq> {0} \<and> f ` S \<subseteq> {1}" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1018 |
using assms unfolding normal_space_iff_Urysohn_alt |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1019 |
by (metis Diff_iff \<open>closedin X S\<close> disjnt_iff subsetD) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1020 |
then show "\<exists>f. continuous_map X euclideanreal f \<and> f x = 0 \<and> f ` S \<subseteq> {1}" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1021 |
by (meson \<open>U \<subseteq> C\<close> \<open>x \<in> U\<close> image_subset_iff singletonD subsetD) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1022 |
qed |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1023 |
|
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1024 |
lemma normal_imp_completely_regular_space_gen: |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1025 |
"\<lbrakk>normal_space X; t1_space X \<or> Hausdorff_space X \<or> regular_space X\<rbrakk> \<Longrightarrow> completely_regular_space X" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1026 |
using normal_imp_completely_regular_space_A normal_imp_completely_regular_space_B t1_or_Hausdorff_space by blast |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1027 |
|
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1028 |
lemma normal_imp_completely_regular_space: |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1029 |
"\<lbrakk>normal_space X; Hausdorff_space X \<or> regular_space X\<rbrakk> \<Longrightarrow> completely_regular_space X" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1030 |
by (simp add: normal_imp_completely_regular_space_gen) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1031 |
|
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1032 |
lemma (in Metric_space) completely_regular_space_mtopology: |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1033 |
"completely_regular_space mtopology" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1034 |
by (simp add: normal_imp_completely_regular_space normal_space_mtopology regular_space_mtopology) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1035 |
|
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1036 |
lemma metrizable_imp_completely_regular_space: |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1037 |
"metrizable_space X \<Longrightarrow> completely_regular_space X" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1038 |
by (simp add: metrizable_imp_normal_space metrizable_imp_regular_space normal_imp_completely_regular_space) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1039 |
|
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1040 |
lemma completely_regular_space_discrete_topology: |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1041 |
"completely_regular_space(discrete_topology U)" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1042 |
by (simp add: normal_imp_completely_regular_space normal_space_discrete_topology) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1043 |
|
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1044 |
lemma completely_regular_space_subtopology: |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1045 |
assumes "completely_regular_space X" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1046 |
shows "completely_regular_space (subtopology X S)" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1047 |
unfolding completely_regular_space_def |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1048 |
proof clarify |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1049 |
fix A x |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1050 |
assume "closedin (subtopology X S) A" and x: "x \<in> topspace (subtopology X S)" and "x \<notin> A" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1051 |
then obtain T where "closedin X T" "A = S \<inter> T" "x \<in> topspace X" "x \<in> S" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1052 |
by (force simp: closedin_subtopology_alt image_iff) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1053 |
then show "\<exists>f. continuous_map (subtopology X S) (top_of_set {0::real..1}) f \<and> f x = 0 \<and> f ` A \<subseteq> {1}" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1054 |
using assms \<open>x \<notin> A\<close> |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1055 |
apply (simp add: completely_regular_space_def continuous_map_from_subtopology) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1056 |
using continuous_map_from_subtopology by fastforce |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1057 |
qed |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1058 |
|
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1059 |
lemma completely_regular_space_retraction_map_image: |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1060 |
" \<lbrakk>retraction_map X Y r; completely_regular_space X\<rbrakk> \<Longrightarrow> completely_regular_space Y" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1061 |
using completely_regular_space_subtopology hereditary_imp_retractive_property homeomorphic_completely_regular_space by blast |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1062 |
|
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1063 |
lemma completely_regular_imp_regular_space: |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1064 |
assumes "completely_regular_space X" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1065 |
shows "regular_space X" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1066 |
proof - |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1067 |
have *: "\<exists>U V. openin X U \<and> openin X V \<and> a \<in> U \<and> C \<subseteq> V \<and> disjnt U V" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1068 |
if contf: "continuous_map X euclideanreal f" and a: "a \<in> topspace X - C" and "closedin X C" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1069 |
and fim: "f ` topspace X \<subseteq> {0..1}" and f0: "f a = 0" and f1: "f ` C \<subseteq> {1}" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1070 |
for C a f |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1071 |
proof (intro exI conjI) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1072 |
show "openin X {x \<in> topspace X. f x \<in> {..<1 / 2}}" "openin X {x \<in> topspace X. f x \<in> {1 / 2<..}}" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1073 |
using openin_continuous_map_preimage [OF contf] |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1074 |
by (meson open_lessThan open_greaterThan open_openin)+ |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1075 |
show "a \<in> {x \<in> topspace X. f x \<in> {..<1 / 2}}" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1076 |
using a f0 by auto |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1077 |
show "C \<subseteq> {x \<in> topspace X. f x \<in> {1 / 2<..}}" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1078 |
using \<open>closedin X C\<close> f1 closedin_subset by auto |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1079 |
qed (auto simp: disjnt_iff) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1080 |
show ?thesis |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1081 |
using assms |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1082 |
unfolding completely_regular_space_def regular_space_def continuous_map_in_subtopology |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1083 |
by (meson "*") |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1084 |
qed |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1085 |
|
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1086 |
|
78274
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
1087 |
proposition locally_compact_regular_imp_completely_regular_space: |
78127
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1088 |
assumes "locally_compact_space X" "Hausdorff_space X \<or> regular_space X" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1089 |
shows "completely_regular_space X" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1090 |
unfolding completely_regular_space_def |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1091 |
proof clarify |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1092 |
fix S x |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1093 |
assume "closedin X S" and "x \<in> topspace X" and "x \<notin> S" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1094 |
have "regular_space X" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1095 |
using assms locally_compact_Hausdorff_imp_regular_space by blast |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1096 |
then have nbase: "neighbourhood_base_of (\<lambda>C. compactin X C \<and> closedin X C) X" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1097 |
using assms(1) locally_compact_regular_space_neighbourhood_base by blast |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1098 |
then obtain U M where "openin X U" "compactin X M" "closedin X M" "x \<in> U" "U \<subseteq> M" "M \<subseteq> topspace X - S" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1099 |
unfolding neighbourhood_base_of by (metis (no_types, lifting) Diff_iff \<open>closedin X S\<close> \<open>x \<in> topspace X\<close> \<open>x \<notin> S\<close> closedin_def) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1100 |
then have "M \<subseteq> topspace X" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1101 |
by blast |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1102 |
obtain V K where "openin X V" "closedin X K" "x \<in> V" "V \<subseteq> K" "K \<subseteq> U" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1103 |
by (metis (no_types, lifting) \<open>openin X U\<close> \<open>x \<in> U\<close> neighbourhood_base_of nbase) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1104 |
have "compact_space (subtopology X M)" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1105 |
by (simp add: \<open>compactin X M\<close> compact_space_subtopology) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1106 |
then have "normal_space (subtopology X M)" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1107 |
by (simp add: \<open>regular_space X\<close> compact_Hausdorff_or_regular_imp_normal_space regular_space_subtopology) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1108 |
moreover have "closedin (subtopology X M) K" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1109 |
using \<open>K \<subseteq> U\<close> \<open>U \<subseteq> M\<close> \<open>closedin X K\<close> closedin_subset_topspace by fastforce |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1110 |
moreover have "closedin (subtopology X M) (M - U)" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1111 |
by (simp add: \<open>closedin X M\<close> \<open>openin X U\<close> closedin_diff closedin_subset_topspace) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1112 |
moreover have "disjnt K (M - U)" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1113 |
by (meson DiffD2 \<open>K \<subseteq> U\<close> disjnt_iff subsetD) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1114 |
ultimately obtain f::"'a\<Rightarrow>real" where contf: "continuous_map (subtopology X M) (top_of_set {0..1}) f" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1115 |
and f0: "f ` K \<subseteq> {0}" and f1: "f ` (M - U) \<subseteq> {1}" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1116 |
using Urysohn_lemma [of "subtopology X M" K "M-U" 0 1] by auto |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1117 |
then obtain g::"'a\<Rightarrow>real" where contg: "continuous_map (subtopology X M) euclidean g" and gim: "g ` M \<subseteq> {0..1}" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1118 |
and g0: "\<And>x. x \<in> K \<Longrightarrow> g x = 0" and g1: "\<And>x. \<lbrakk>x \<in> M; x \<notin> U\<rbrakk> \<Longrightarrow> g x = 1" |
78202 | 1119 |
using \<open>M \<subseteq> topspace X\<close> by (force simp: continuous_map_in_subtopology image_subset_iff) |
78127
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1120 |
show "\<exists>f::'a\<Rightarrow>real. continuous_map X (top_of_set {0..1}) f \<and> f x = 0 \<and> f ` S \<subseteq> {1}" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1121 |
proof (intro exI conjI) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1122 |
show "continuous_map X (top_of_set {0..1}) (\<lambda>x. if x \<in> M then g x else 1)" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1123 |
unfolding continuous_map_closedin |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1124 |
proof (intro strip conjI) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1125 |
fix C |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1126 |
assume C: "closedin (top_of_set {0::real..1}) C" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1127 |
have eq: "{x \<in> topspace X. (if x \<in> M then g x else 1) \<in> C} = {x \<in> M. g x \<in> C} \<union> (if 1 \<in> C then topspace X - U else {})" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1128 |
using \<open>U \<subseteq> M\<close> \<open>M \<subseteq> topspace X\<close> g1 by auto |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1129 |
show "closedin X {x \<in> topspace X. (if x \<in> M then g x else 1) \<in> C}" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1130 |
unfolding eq |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1131 |
proof (intro closedin_Un) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1132 |
have "closedin euclidean C" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1133 |
using C closed_closedin closedin_closed_trans by blast |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1134 |
then have "closedin (subtopology X M) {x \<in> M. g x \<in> C}" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1135 |
using closedin_continuous_map_preimage_gen [OF contg] \<open>M \<subseteq> topspace X\<close> by auto |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1136 |
then show "closedin X {x \<in> M. g x \<in> C}" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1137 |
using \<open>closedin X M\<close> closedin_trans_full by blast |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1138 |
qed (use \<open>openin X U\<close> in force) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1139 |
qed (use gim in force) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1140 |
show "(if x \<in> M then g x else 1) = 0" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1141 |
using \<open>U \<subseteq> M\<close> \<open>V \<subseteq> K\<close> g0 \<open>x \<in> U\<close> \<open>x \<in> V\<close> by auto |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1142 |
show "(\<lambda>x. if x \<in> M then g x else 1) ` S \<subseteq> {1}" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1143 |
using \<open>M \<subseteq> topspace X - S\<close> by auto |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1144 |
qed |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1145 |
qed |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1146 |
|
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1147 |
lemma completely_regular_eq_regular_space: |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1148 |
"locally_compact_space X |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1149 |
\<Longrightarrow> (completely_regular_space X \<longleftrightarrow> regular_space X)" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1150 |
using completely_regular_imp_regular_space locally_compact_regular_imp_completely_regular_space |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1151 |
by blast |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1152 |
|
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1153 |
lemma completely_regular_space_prod_topology: |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1154 |
"completely_regular_space (prod_topology X Y) \<longleftrightarrow> |
78336 | 1155 |
(prod_topology X Y) = trivial_topology \<or> |
78127
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1156 |
completely_regular_space X \<and> completely_regular_space Y" (is "?lhs=?rhs") |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1157 |
proof |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1158 |
assume ?lhs then show ?rhs |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1159 |
by (rule topological_property_of_prod_component) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1160 |
(auto simp: completely_regular_space_subtopology homeomorphic_completely_regular_space) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1161 |
next |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1162 |
assume R: ?rhs |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1163 |
show ?lhs |
78336 | 1164 |
proof (cases "(prod_topology X Y) = trivial_topology") |
78127
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1165 |
case False |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1166 |
then have X: "completely_regular_space X" and Y: "completely_regular_space Y" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1167 |
using R by blast+ |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1168 |
show ?thesis |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1169 |
unfolding completely_regular_space_alt' |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1170 |
proof clarify |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1171 |
fix W x y |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1172 |
assume "openin (prod_topology X Y) W" and "(x, y) \<in> W" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1173 |
then obtain U V where "openin X U" "openin Y V" "x \<in> U" "y \<in> V" "U\<times>V \<subseteq> W" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1174 |
by (force simp: openin_prod_topology_alt) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1175 |
then have "x \<in> topspace X" "y \<in> topspace Y" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1176 |
using openin_subset by fastforce+ |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1177 |
obtain f where contf: "continuous_map X euclideanreal f" and "f x = 0" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1178 |
and f1: "\<And>x. x \<in> topspace X \<Longrightarrow> x \<notin> U \<Longrightarrow> f x = 1" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1179 |
using X \<open>openin X U\<close> \<open>x \<in> U\<close> unfolding completely_regular_space_alt' |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1180 |
by (smt (verit, best) Diff_iff image_subset_iff singletonD) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1181 |
obtain g where contg: "continuous_map Y euclideanreal g" and "g y = 0" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1182 |
and g1: "\<And>y. y \<in> topspace Y \<Longrightarrow> y \<notin> V \<Longrightarrow> g y = 1" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1183 |
using Y \<open>openin Y V\<close> \<open>y \<in> V\<close> unfolding completely_regular_space_alt' |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1184 |
by (smt (verit, best) Diff_iff image_subset_iff singletonD) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1185 |
define h where "h \<equiv> \<lambda>(x,y). 1 - (1 - f x) * (1 - g y)" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1186 |
show "\<exists>h. continuous_map (prod_topology X Y) euclideanreal h \<and> h (x,y) = 0 \<and> h ` (topspace (prod_topology X Y) - W) \<subseteq> {1}" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1187 |
proof (intro exI conjI) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1188 |
have "continuous_map (prod_topology X Y) euclideanreal (f \<circ> fst)" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1189 |
using contf continuous_map_of_fst by blast |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1190 |
moreover |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1191 |
have "continuous_map (prod_topology X Y) euclideanreal (g \<circ> snd)" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1192 |
using contg continuous_map_of_snd by blast |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1193 |
ultimately |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1194 |
show "continuous_map (prod_topology X Y) euclideanreal h" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1195 |
unfolding o_def h_def case_prod_unfold |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1196 |
by (intro continuous_intros) auto |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1197 |
show "h (x, y) = 0" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1198 |
by (simp add: h_def \<open>f x = 0\<close> \<open>g y = 0\<close>) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1199 |
show "h ` (topspace (prod_topology X Y) - W) \<subseteq> {1}" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1200 |
using \<open>U \<times> V \<subseteq> W\<close> f1 g1 by (force simp: h_def) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1201 |
qed |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1202 |
qed |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1203 |
qed (force simp: completely_regular_space_def) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1204 |
qed |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1205 |
|
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1206 |
|
78274
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
1207 |
proposition completely_regular_space_product_topology: |
78127
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1208 |
"completely_regular_space (product_topology X I) \<longleftrightarrow> |
78336 | 1209 |
(\<exists>i\<in>I. X i = trivial_topology) \<or> (\<forall>i \<in> I. completely_regular_space (X i))" |
78127
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1210 |
(is "?lhs \<longleftrightarrow> ?rhs") |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1211 |
proof |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1212 |
assume ?lhs then show ?rhs |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1213 |
by (rule topological_property_of_product_component) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1214 |
(auto simp: completely_regular_space_subtopology homeomorphic_completely_regular_space) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1215 |
next |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1216 |
assume R: ?rhs |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1217 |
show ?lhs |
78336 | 1218 |
proof (cases "\<exists>i\<in>I. X i = trivial_topology") |
78127
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1219 |
case False |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1220 |
show ?thesis |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1221 |
unfolding completely_regular_space_alt' |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1222 |
proof clarify |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1223 |
fix W x |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1224 |
assume W: "openin (product_topology X I) W" and "x \<in> W" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1225 |
then obtain U where finU: "finite {i \<in> I. U i \<noteq> topspace (X i)}" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1226 |
and ope: "\<And>i. i\<in>I \<Longrightarrow> openin (X i) (U i)" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1227 |
and x: "x \<in> Pi\<^sub>E I U" and "Pi\<^sub>E I U \<subseteq> W" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1228 |
by (auto simp: openin_product_topology_alt) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1229 |
have "\<forall>i \<in> I. openin (X i) (U i) \<and> x i \<in> U i |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1230 |
\<longrightarrow> (\<exists>f. continuous_map (X i) euclideanreal f \<and> |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1231 |
f (x i) = 0 \<and> (\<forall>x \<in> topspace(X i). x \<notin> U i \<longrightarrow> f x = 1))" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1232 |
using R unfolding completely_regular_space_alt' |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1233 |
by (smt (verit) DiffI False image_subset_iff singletonD) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1234 |
with ope x have "\<And>i. \<exists>f. i \<in> I \<longrightarrow> continuous_map (X i) euclideanreal f \<and> |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1235 |
f (x i) = 0 \<and> (\<forall>x \<in> topspace (X i). x \<notin> U i \<longrightarrow> f x = 1)" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1236 |
by auto |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1237 |
then obtain f where f: "\<And>i. i \<in> I \<Longrightarrow> continuous_map (X i) euclideanreal (f i) \<and> |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1238 |
f i (x i) = 0 \<and> (\<forall>x \<in> topspace (X i). x \<notin> U i \<longrightarrow> f i x = 1)" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1239 |
by metis |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1240 |
define h where "h \<equiv> \<lambda>z. 1 - prod (\<lambda>i. 1 - f i (z i)) {i\<in>I. U i \<noteq> topspace(X i)}" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1241 |
show "\<exists>h. continuous_map (product_topology X I) euclideanreal h \<and> h x = 0 \<and> |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1242 |
h ` (topspace (product_topology X I) - W) \<subseteq> {1}" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1243 |
proof (intro conjI exI) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1244 |
have "continuous_map (product_topology X I) euclidean (f i \<circ> (\<lambda>x. x i))" if "i\<in>I" for i |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1245 |
using f that |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1246 |
by (blast intro: continuous_intros continuous_map_product_projection) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1247 |
then show "continuous_map (product_topology X I) euclideanreal h" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1248 |
unfolding h_def o_def by (intro continuous_intros) (auto simp: finU) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1249 |
show "h x = 0" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1250 |
by (simp add: h_def f) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1251 |
show "h ` (topspace (product_topology X I) - W) \<subseteq> {1}" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1252 |
proof - |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1253 |
have "\<exists>i. i \<in> I \<and> U i \<noteq> topspace (X i) \<and> f i (x' i) = 1" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1254 |
if "x' \<in> (\<Pi>\<^sub>E i\<in>I. topspace (X i))" "x' \<notin> W" for x' |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1255 |
using that \<open>Pi\<^sub>E I U \<subseteq> W\<close> by (smt (verit, best) PiE_iff f in_mono) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1256 |
then show ?thesis |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1257 |
by (auto simp: f h_def finU) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1258 |
qed |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1259 |
qed |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1260 |
qed |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1261 |
qed (force simp: completely_regular_space_def) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1262 |
qed |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1263 |
|
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1264 |
|
78250
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
1265 |
lemma zero_dimensional_imp_completely_regular_space: |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
1266 |
assumes "X dim_le 0" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
1267 |
shows "completely_regular_space X" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
1268 |
proof (clarsimp simp: completely_regular_space_def) |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
1269 |
fix C a |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
1270 |
assume "closedin X C" and "a \<in> topspace X" and "a \<notin> C" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
1271 |
then obtain U where "closedin X U" "openin X U" "a \<in> U" and U: "U \<subseteq> topspace X - C" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
1272 |
using assms by (force simp add: closedin_def dimension_le_0_neighbourhood_base_of_clopen open_neighbourhood_base_of) |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
1273 |
show "\<exists>f. continuous_map X (top_of_set {0::real..1}) f \<and> f a = 0 \<and> f ` C \<subseteq> {1}" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
1274 |
proof (intro exI conjI) |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
1275 |
have "continuous_map X euclideanreal (\<lambda>x. if x \<in> U then 0 else 1)" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
1276 |
using \<open>closedin X U\<close> \<open>openin X U\<close> apply (clarsimp simp: continuous_map_def closedin_def) |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
1277 |
by (smt (verit) Diff_iff mem_Collect_eq openin_subopen subset_eq) |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
1278 |
then show "continuous_map X (top_of_set {0::real..1}) (\<lambda>x. if x \<in> U then 0 else 1)" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
1279 |
by (auto simp: continuous_map_in_subtopology) |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
1280 |
qed (use U \<open>a \<in> U\<close> in auto) |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
1281 |
qed |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
1282 |
|
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
1283 |
lemma zero_dimensional_imp_regular_space: "X dim_le 0 \<Longrightarrow> regular_space X" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
1284 |
by (simp add: completely_regular_imp_regular_space zero_dimensional_imp_completely_regular_space) |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
1285 |
|
78127
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1286 |
lemma (in Metric_space) t1_space_mtopology: |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1287 |
"t1_space mtopology" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1288 |
using Hausdorff_space_mtopology t1_or_Hausdorff_space by blast |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1289 |
|
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1290 |
|
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1291 |
subsection\<open>More generally, the k-ification functor\<close> |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1292 |
|
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1293 |
definition kification_open |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1294 |
where "kification_open \<equiv> |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1295 |
\<lambda>X S. S \<subseteq> topspace X \<and> (\<forall>K. compactin X K \<longrightarrow> openin (subtopology X K) (K \<inter> S))" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1296 |
|
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1297 |
definition kification |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1298 |
where "kification X \<equiv> topology (kification_open X)" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1299 |
|
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1300 |
lemma istopology_kification_open: "istopology (kification_open X)" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1301 |
unfolding istopology_def |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1302 |
proof (intro conjI strip) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1303 |
show "kification_open X (S \<inter> T)" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1304 |
if "kification_open X S" and "kification_open X T" for S T |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1305 |
using that unfolding kification_open_def |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1306 |
by (smt (verit, best) inf.idem inf_commute inf_left_commute le_infI2 openin_Int) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1307 |
show "kification_open X (\<Union> \<K>)" if "\<forall>K\<in>\<K>. kification_open X K" for \<K> |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1308 |
using that unfolding kification_open_def Int_Union by blast |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1309 |
qed |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1310 |
|
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1311 |
lemma openin_kification: |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1312 |
"openin (kification X) U \<longleftrightarrow> |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1313 |
U \<subseteq> topspace X \<and> |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1314 |
(\<forall>K. compactin X K \<longrightarrow> openin (subtopology X K) (K \<inter> U))" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1315 |
by (metis topology_inverse' kification_def istopology_kification_open kification_open_def) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1316 |
|
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1317 |
lemma openin_kification_finer: |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1318 |
"openin X S \<Longrightarrow> openin (kification X) S" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1319 |
by (simp add: openin_kification openin_subset openin_subtopology_Int2) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1320 |
|
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1321 |
lemma topspace_kification [simp]: |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1322 |
"topspace(kification X) = topspace X" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1323 |
by (meson openin_kification openin_kification_finer openin_topspace subset_antisym) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1324 |
|
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1325 |
lemma closedin_kification: |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1326 |
"closedin (kification X) U \<longleftrightarrow> |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1327 |
U \<subseteq> topspace X \<and> |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1328 |
(\<forall>K. compactin X K \<longrightarrow> closedin (subtopology X K) (K \<inter> U))" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1329 |
proof (cases "U \<subseteq> topspace X") |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1330 |
case True |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1331 |
then show ?thesis |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1332 |
by (simp add: closedin_def Diff_Int_distrib inf_commute le_infI2 openin_kification) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1333 |
qed (simp add: closedin_def) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1334 |
|
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1335 |
lemma closedin_kification_finer: "closedin X S \<Longrightarrow> closedin (kification X) S" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1336 |
by (simp add: closedin_def openin_kification_finer) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1337 |
|
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1338 |
lemma kification_eq_self: "kification X = X \<longleftrightarrow> k_space X" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1339 |
unfolding fun_eq_iff openin_kification k_space_alt openin_inject [symmetric] |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1340 |
by (metis openin_closedin_eq) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1341 |
|
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1342 |
lemma compactin_kification [simp]: |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1343 |
"compactin (kification X) K \<longleftrightarrow> compactin X K" (is "?lhs \<longleftrightarrow> ?rhs") |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1344 |
proof |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1345 |
assume ?lhs then show ?rhs |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1346 |
by (simp add: compactin_contractive openin_kification_finer) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1347 |
next |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1348 |
assume R: ?rhs |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1349 |
show ?lhs |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1350 |
unfolding compactin_def |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1351 |
proof (intro conjI strip) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1352 |
show "K \<subseteq> topspace (kification X)" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1353 |
by (simp add: R compactin_subset_topspace) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1354 |
fix \<U> |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1355 |
assume \<U>: "Ball \<U> (openin (kification X)) \<and> K \<subseteq> \<Union> \<U>" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1356 |
then have *: "\<And>U. U \<in> \<U> \<Longrightarrow> U \<subseteq> topspace X \<and> openin (subtopology X K) (K \<inter> U)" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1357 |
by (simp add: R openin_kification) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1358 |
have "K \<subseteq> topspace X" "compact_space (subtopology X K)" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1359 |
using R compactin_subspace by force+ |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1360 |
then have "\<exists>V. finite V \<and> V \<subseteq> (\<lambda>U. K \<inter> U) ` \<U> \<and> \<Union> V = topspace (subtopology X K)" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1361 |
unfolding compact_space |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1362 |
by (smt (verit, del_insts) Int_Union \<U> * image_iff inf.order_iff inf_commute topspace_subtopology) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1363 |
then show "\<exists>\<F>. finite \<F> \<and> \<F> \<subseteq> \<U> \<and> K \<subseteq> \<Union> \<F>" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1364 |
by (metis Int_Union \<open>K \<subseteq> topspace X\<close> finite_subset_image inf.orderI topspace_subtopology_subset) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1365 |
qed |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1366 |
qed |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1367 |
|
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1368 |
lemma compact_space_kification [simp]: |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1369 |
"compact_space(kification X) \<longleftrightarrow> compact_space X" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1370 |
by (simp add: compact_space_def) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1371 |
|
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1372 |
lemma kification_kification [simp]: |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1373 |
"kification(kification X) = kification X" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1374 |
unfolding openin_inject [symmetric] |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1375 |
proof |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1376 |
fix U |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1377 |
show "openin (kification (kification X)) U = openin (kification X) U" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1378 |
proof |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1379 |
show "openin (kification (kification X)) U \<Longrightarrow> openin (kification X) U" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1380 |
by (metis compactin_kification inf_commute openin_kification openin_subtopology topspace_kification) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1381 |
qed (simp add: openin_kification_finer) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1382 |
qed |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1383 |
|
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1384 |
lemma k_space_kification [iff]: "k_space(kification X)" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1385 |
using kification_eq_self by fastforce |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1386 |
|
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1387 |
lemma continuous_map_into_kification: |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1388 |
assumes "k_space X" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1389 |
shows "continuous_map X (kification Y) f \<longleftrightarrow> continuous_map X Y f" (is "?lhs \<longleftrightarrow> ?rhs") |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1390 |
proof |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1391 |
assume ?lhs then show ?rhs |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1392 |
by (simp add: continuous_map_def openin_kification_finer) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1393 |
next |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1394 |
assume R: ?rhs |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1395 |
have "openin X {x \<in> topspace X. f x \<in> V}" if V: "openin (kification Y) V" for V |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1396 |
proof - |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1397 |
have "openin (subtopology X K) (K \<inter> {x \<in> topspace X. f x \<in> V})" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1398 |
if "compactin X K" for K |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1399 |
proof - |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1400 |
have "compactin Y (f ` K)" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1401 |
using R image_compactin that by blast |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1402 |
then have "openin (subtopology Y (f ` K)) (f ` K \<inter> V)" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1403 |
by (meson V openin_kification) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1404 |
then obtain U where U: "openin Y U" "f`K \<inter> V = U \<inter> f`K" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1405 |
by (meson openin_subtopology) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1406 |
show ?thesis |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1407 |
unfolding openin_subtopology |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1408 |
proof (intro conjI exI) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1409 |
show "openin X {x \<in> topspace X. f x \<in> U}" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1410 |
using R U openin_continuous_map_preimage_gen by (simp add: continuous_map_def) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1411 |
qed (use U in auto) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1412 |
qed |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1413 |
then show ?thesis |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1414 |
by (metis (full_types) Collect_subset assms k_space_open) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1415 |
qed |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1416 |
with R show ?lhs |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1417 |
by (simp add: continuous_map_def) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1418 |
qed |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1419 |
|
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1420 |
lemma continuous_map_from_kification: |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1421 |
"continuous_map X Y f \<Longrightarrow> continuous_map (kification X) Y f" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1422 |
by (simp add: continuous_map_openin_preimage_eq openin_kification_finer) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1423 |
|
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1424 |
lemma continuous_map_kification: |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1425 |
"continuous_map X Y f \<Longrightarrow> continuous_map (kification X) (kification Y) f" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1426 |
by (simp add: continuous_map_from_kification continuous_map_into_kification) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1427 |
|
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1428 |
lemma subtopology_kification_compact: |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1429 |
assumes "compactin X K" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1430 |
shows "subtopology (kification X) K = subtopology X K" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1431 |
unfolding openin_inject [symmetric] |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1432 |
proof |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1433 |
fix U |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1434 |
show "openin (subtopology (kification X) K) U = openin (subtopology X K) U" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1435 |
by (metis assms inf_commute openin_kification openin_subset openin_subtopology) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1436 |
qed |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1437 |
|
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1438 |
|
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1439 |
lemma subtopology_kification_finer: |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1440 |
assumes "openin (subtopology (kification X) S) U" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1441 |
shows "openin (kification (subtopology X S)) U" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1442 |
using assms |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1443 |
by (fastforce simp: openin_subtopology_alt image_iff openin_kification subtopology_subtopology compactin_subtopology) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1444 |
|
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1445 |
lemma proper_map_from_kification: |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1446 |
assumes "k_space Y" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1447 |
shows "proper_map (kification X) Y f \<longleftrightarrow> proper_map X Y f" (is "?lhs \<longleftrightarrow> ?rhs") |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1448 |
proof |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1449 |
assume ?lhs then show ?rhs |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1450 |
by (simp add: closed_map_def closedin_kification_finer proper_map_alt) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1451 |
next |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1452 |
assume R: ?rhs |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1453 |
have "compactin Y K \<Longrightarrow> compactin X {x \<in> topspace X. f x \<in> K}" for K |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1454 |
using R proper_map_alt by auto |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1455 |
with R show ?lhs |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1456 |
by (simp add: assms proper_map_into_k_space_eq subtopology_kification_compact) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1457 |
qed |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1458 |
|
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1459 |
lemma perfect_map_from_kification: |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1460 |
"\<lbrakk>k_space Y; perfect_map X Y f\<rbrakk> \<Longrightarrow> perfect_map(kification X) Y f" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1461 |
by (simp add: continuous_map_from_kification perfect_map_def proper_map_from_kification) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1462 |
|
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1463 |
lemma k_space_perfect_map_image_eq: |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1464 |
assumes "Hausdorff_space X" "perfect_map X Y f" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1465 |
shows "k_space X \<longleftrightarrow> k_space Y" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1466 |
proof |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1467 |
show "k_space X \<Longrightarrow> k_space Y" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1468 |
using k_space_perfect_map_image assms by blast |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1469 |
assume "k_space Y" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1470 |
have "homeomorphic_map (kification X) X id" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1471 |
unfolding homeomorphic_eq_injective_perfect_map |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1472 |
proof (intro conjI perfect_map_from_composition_right [where f = id]) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1473 |
show "perfect_map (kification X) Y (f \<circ> id)" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1474 |
by (simp add: \<open>k_space Y\<close> assms(2) perfect_map_from_kification) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1475 |
show "continuous_map (kification X) X id" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1476 |
by (simp add: continuous_map_from_kification) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1477 |
qed (use assms perfect_map_def in auto) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1478 |
then show "k_space X" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1479 |
using homeomorphic_k_space homeomorphic_space by blast |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1480 |
qed |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1481 |
|
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1482 |
|
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1483 |
subsection\<open>One-point compactifications and the Alexandroff extension construction\<close> |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1484 |
|
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1485 |
lemma one_point_compactification_dense: |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1486 |
"\<lbrakk>compact_space X; \<not> compactin X (topspace X - {a})\<rbrakk> \<Longrightarrow> X closure_of (topspace X - {a}) = topspace X" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1487 |
unfolding closure_of_complement |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1488 |
by (metis Diff_empty closedin_compact_space interior_of_eq_empty openin_closedin_eq subset_singletonD) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1489 |
|
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1490 |
lemma one_point_compactification_interior: |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1491 |
"\<lbrakk>compact_space X; \<not> compactin X (topspace X - {a})\<rbrakk> \<Longrightarrow> X interior_of {a} = {}" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1492 |
by (simp add: interior_of_eq_empty_complement one_point_compactification_dense) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1493 |
|
78274
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
1494 |
proposition kc_space_one_point_compactification_gen: |
78127
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1495 |
assumes "compact_space X" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1496 |
shows "kc_space X \<longleftrightarrow> |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1497 |
openin X (topspace X - {a}) \<and> (\<forall>K. compactin X K \<and> a\<notin>K \<longrightarrow> closedin X K) \<and> |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1498 |
k_space (subtopology X (topspace X - {a})) \<and> kc_space (subtopology X (topspace X - {a}))" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1499 |
(is "?lhs \<longleftrightarrow> ?rhs") |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1500 |
proof |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1501 |
assume L: ?lhs show ?rhs |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1502 |
proof (intro conjI strip) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1503 |
show "openin X (topspace X - {a})" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1504 |
using L kc_imp_t1_space t1_space_openin_delete_alt by auto |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1505 |
then show "k_space (subtopology X (topspace X - {a}))" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1506 |
by (simp add: L assms k_space_open_subtopology_aux) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1507 |
show "closedin X k" if "compactin X k \<and> a \<notin> k" for k :: "'a set" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1508 |
using L kc_space_def that by blast |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1509 |
show "kc_space (subtopology X (topspace X - {a}))" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1510 |
by (simp add: L kc_space_subtopology) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1511 |
qed |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1512 |
next |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1513 |
assume R: ?rhs |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1514 |
show ?lhs |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1515 |
unfolding kc_space_def |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1516 |
proof (intro strip) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1517 |
fix S |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1518 |
assume "compactin X S" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1519 |
then have "S \<subseteq>topspace X" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1520 |
by (simp add: compactin_subset_topspace) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1521 |
show "closedin X S" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1522 |
proof (cases "a \<in> S") |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1523 |
case True |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1524 |
then have "topspace X - S = topspace X - {a} - (S - {a})" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1525 |
by auto |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1526 |
moreover have "openin X (topspace X - {a} - (S - {a}))" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1527 |
proof (rule openin_trans_full) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1528 |
show "openin (subtopology X (topspace X - {a})) (topspace X - {a} - (S - {a}))" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1529 |
proof |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1530 |
show "openin (subtopology X (topspace X - {a})) (topspace X - {a})" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1531 |
using R openin_open_subtopology by blast |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1532 |
have "closedin (subtopology X ((topspace X - {a}) \<inter> K)) (K \<inter> (S - {a}))" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1533 |
if "compactin X K" "K \<subseteq> topspace X - {a}" for K |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1534 |
proof (intro closedin_subset_topspace) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1535 |
show "closedin X (K \<inter> (S - {a}))" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1536 |
using that |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1537 |
by (metis IntD1 Int_insert_right_if0 R True \<open>compactin X S\<close> closed_Int_compactin insert_Diff subset_Diff_insert) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1538 |
qed (use that in auto) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1539 |
moreover have "k_space (subtopology X (topspace X - {a}))" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1540 |
using R by blast |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1541 |
moreover have "S-{a} \<subseteq> topspace X \<and> S-{a} \<subseteq> topspace X - {a}" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1542 |
using \<open>S \<subseteq> topspace X\<close> by auto |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1543 |
ultimately show "closedin (subtopology X (topspace X - {a})) (S - {a})" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1544 |
using \<open>S \<subseteq> topspace X\<close> True |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1545 |
by (simp add: k_space_def compactin_subtopology subtopology_subtopology) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1546 |
qed |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1547 |
show "openin X (topspace X - {a})" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1548 |
by (simp add: R) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1549 |
qed |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1550 |
ultimately show ?thesis |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1551 |
by (simp add: \<open>S \<subseteq> topspace X\<close> closedin_def) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1552 |
next |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1553 |
case False |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1554 |
then show ?thesis |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1555 |
by (simp add: R \<open>compactin X S\<close>) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1556 |
qed |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1557 |
qed |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1558 |
qed |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1559 |
|
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1560 |
|
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1561 |
inductive Alexandroff_open for X where |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1562 |
base: "openin X U \<Longrightarrow> Alexandroff_open X (Some ` U)" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1563 |
| ext: "\<lbrakk>compactin X C; closedin X C\<rbrakk> \<Longrightarrow> Alexandroff_open X (insert None (Some ` (topspace X - C)))" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1564 |
|
78128
3d2db8057b9f
Hiding the constructor names, particularly to avoid conflicts involving "ext"
paulson <lp15@cam.ac.uk>
parents:
78127
diff
changeset
|
1565 |
hide_fact (open) base ext |
3d2db8057b9f
Hiding the constructor names, particularly to avoid conflicts involving "ext"
paulson <lp15@cam.ac.uk>
parents:
78127
diff
changeset
|
1566 |
|
78127
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1567 |
lemma Alexandroff_open_iff: "Alexandroff_open X S \<longleftrightarrow> |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1568 |
(\<exists>U. (S = Some ` U \<and> openin X U) \<or> (S = insert None (Some ` (topspace X - U)) \<and> compactin X U \<and> closedin X U))" |
78128
3d2db8057b9f
Hiding the constructor names, particularly to avoid conflicts involving "ext"
paulson <lp15@cam.ac.uk>
parents:
78127
diff
changeset
|
1569 |
by (meson Alexandroff_open.cases Alexandroff_open.ext Alexandroff_open.base) |
78127
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1570 |
|
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1571 |
lemma Alexandroff_open_Un_aux: |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1572 |
assumes U: "openin X U" and "Alexandroff_open X T" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1573 |
shows "Alexandroff_open X (Some ` U \<union> T)" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1574 |
using \<open>Alexandroff_open X T\<close> |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1575 |
proof (induction rule: Alexandroff_open.induct) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1576 |
case (base V) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1577 |
then show ?case |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1578 |
by (metis Alexandroff_open.base U image_Un openin_Un) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1579 |
next |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1580 |
case (ext C) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1581 |
have "U \<subseteq> topspace X" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1582 |
by (simp add: U openin_subset) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1583 |
then have eq: "Some ` U \<union> insert None (Some ` (topspace X - C)) = insert None (Some ` (topspace X - (C \<inter> (topspace X - U))))" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1584 |
by force |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1585 |
have "closedin X (C \<inter> (topspace X - U))" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1586 |
using U ext.hyps(2) by blast |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1587 |
moreover |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1588 |
have "compactin X (C \<inter> (topspace X - U))" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1589 |
using U compact_Int_closedin ext.hyps(1) by blast |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1590 |
ultimately show ?case |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1591 |
unfolding eq using Alexandroff_open.ext by blast |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1592 |
qed |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1593 |
|
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1594 |
lemma Alexandroff_open_Un: |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1595 |
assumes "Alexandroff_open X S" and "Alexandroff_open X T" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1596 |
shows "Alexandroff_open X (S \<union> T)" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1597 |
using assms |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1598 |
proof (induction rule: Alexandroff_open.induct) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1599 |
case (base U) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1600 |
then show ?case |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1601 |
by (simp add: Alexandroff_open_Un_aux) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1602 |
next |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1603 |
case (ext C) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1604 |
then show ?case |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1605 |
by (smt (verit, best) Alexandroff_open_Un_aux Alexandroff_open_iff Un_commute Un_insert_left closedin_def insert_absorb2) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1606 |
qed |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1607 |
|
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1608 |
lemma Alexandroff_open_Int_aux: |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1609 |
assumes U: "openin X U" and "Alexandroff_open X T" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1610 |
shows "Alexandroff_open X (Some ` U \<inter> T)" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1611 |
using \<open>Alexandroff_open X T\<close> |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1612 |
proof (induction rule: Alexandroff_open.induct) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1613 |
case (base V) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1614 |
then show ?case |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1615 |
by (metis Alexandroff_open.base U image_Int inj_Some openin_Int) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1616 |
next |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1617 |
case (ext C) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1618 |
have eq: "Some ` U \<inter> insert None (Some ` (topspace X - C)) = Some ` (topspace X - (C \<union> (topspace X - U)))" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1619 |
by force |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1620 |
have "openin X (topspace X - (C \<union> (topspace X - U)))" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1621 |
using U ext.hyps(2) by blast |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1622 |
then show ?case |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1623 |
unfolding eq using Alexandroff_open.base by blast |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1624 |
qed |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1625 |
|
78274
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
1626 |
proposition istopology_Alexandroff_open: "istopology (Alexandroff_open X)" |
78127
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1627 |
unfolding istopology_def |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1628 |
proof (intro conjI strip) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1629 |
fix S T |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1630 |
assume "Alexandroff_open X S" and "Alexandroff_open X T" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1631 |
then show "Alexandroff_open X (S \<inter> T)" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1632 |
proof (induction rule: Alexandroff_open.induct) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1633 |
case (base U) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1634 |
then show ?case |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1635 |
using Alexandroff_open_Int_aux by blast |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1636 |
next |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1637 |
case EC: (ext C) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1638 |
show ?case |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1639 |
using \<open>Alexandroff_open X T\<close> |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1640 |
proof (induction rule: Alexandroff_open.induct) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1641 |
case (base V) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1642 |
then show ?case |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1643 |
by (metis Alexandroff_open.ext Alexandroff_open_Int_aux EC.hyps inf_commute) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1644 |
next |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1645 |
case (ext D) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1646 |
have eq: "insert None (Some ` (topspace X - C)) \<inter> insert None (Some ` (topspace X - D)) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1647 |
= insert None (Some ` (topspace X - (C \<union> D)))" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1648 |
by auto |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1649 |
show ?case |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1650 |
unfolding eq |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1651 |
by (simp add: Alexandroff_open.ext EC.hyps closedin_Un compactin_Un ext.hyps) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1652 |
qed |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1653 |
qed |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1654 |
next |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1655 |
fix \<K> |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1656 |
assume \<section>: "\<forall>K\<in>\<K>. Alexandroff_open X K" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1657 |
show "Alexandroff_open X (\<Union>\<K>)" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1658 |
proof (cases "None \<in> \<Union>\<K>") |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1659 |
case True |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1660 |
have "\<forall>K\<in>\<K>. \<exists>U. (openin X U \<and> K = Some ` U) \<or> (K = insert None (Some ` (topspace X - U)) \<and> compactin X U \<and> closedin X U)" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1661 |
by (metis \<section> Alexandroff_open_iff) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1662 |
then obtain U where U: |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1663 |
"\<And>K. K \<in> \<K> \<Longrightarrow> openin X (U K) \<and> K = Some ` (U K) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1664 |
\<or> (K = insert None (Some ` (topspace X - U K)) \<and> compactin X (U K) \<and> closedin X (U K))" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1665 |
by metis |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1666 |
define \<K>N where "\<K>N \<equiv> {K \<in> \<K>. None \<in> K}" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1667 |
define A where "A \<equiv> \<Union>K\<in>\<K>-\<K>N. U K" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1668 |
define B where "B \<equiv> \<Inter>K\<in>\<K>N. U K" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1669 |
have U1: "\<And>K. K \<in> \<K>-\<K>N \<Longrightarrow> openin X (U K) \<and> K = Some ` (U K)" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1670 |
using U \<K>N_def by auto |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1671 |
have U2: "\<And>K. K \<in> \<K>N \<Longrightarrow> K = insert None (Some ` (topspace X - U K)) \<and> compactin X (U K) \<and> closedin X (U K)" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1672 |
using U \<K>N_def by auto |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1673 |
have eqA: "\<Union>(\<K>-\<K>N) = Some ` A" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1674 |
proof |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1675 |
show "\<Union> (\<K> - \<K>N) \<subseteq> Some ` A" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1676 |
by (metis A_def Sup_le_iff U1 UN_upper subset_image_iff) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1677 |
show "Some ` A \<subseteq> \<Union> (\<K> - \<K>N)" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1678 |
using A_def U1 by blast |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1679 |
qed |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1680 |
have eqB: "\<Union>\<K>N = insert None (Some ` (topspace X - B))" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1681 |
using U2 True |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1682 |
by (auto simp: B_def image_iff \<K>N_def) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1683 |
have "\<Union>\<K> = \<Union>\<K>N \<union> \<Union>(\<K>-\<K>N)" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1684 |
by (auto simp: \<K>N_def) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1685 |
then have eq: "\<Union>\<K> = (Some ` A) \<union> (insert None (Some ` (topspace X - B)))" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1686 |
by (simp add: eqA eqB Un_commute) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1687 |
show ?thesis |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1688 |
unfolding eq |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1689 |
proof (intro Alexandroff_open_Un Alexandroff_open.intros) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1690 |
show "openin X A" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1691 |
using A_def U1 by blast |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1692 |
show "closedin X B" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1693 |
unfolding B_def using U2 True \<K>N_def by auto |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1694 |
show "compactin X B" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1695 |
by (metis B_def U2 eqB Inf_lower Union_iff \<open>closedin X B\<close> closed_compactin imageI insertI1) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1696 |
qed |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1697 |
next |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1698 |
case False |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1699 |
then have "\<forall>K\<in>\<K>. \<exists>U. openin X U \<and> K = Some ` U" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1700 |
by (metis Alexandroff_open.simps UnionI \<section> insertCI) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1701 |
then obtain U where U: "\<forall>K\<in>\<K>. openin X (U K) \<and> K = Some ` (U K)" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1702 |
by metis |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1703 |
then have eq: "\<Union>\<K> = Some ` (\<Union> K\<in>\<K>. U K)" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1704 |
using image_iff by fastforce |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1705 |
show ?thesis |
78128
3d2db8057b9f
Hiding the constructor names, particularly to avoid conflicts involving "ext"
paulson <lp15@cam.ac.uk>
parents:
78127
diff
changeset
|
1706 |
unfolding eq by (simp add: U Alexandroff_open.base openin_clauses(3)) |
78127
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1707 |
qed |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1708 |
qed |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1709 |
|
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1710 |
|
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1711 |
definition Alexandroff_compactification where |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1712 |
"Alexandroff_compactification X \<equiv> topology (Alexandroff_open X)" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1713 |
|
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1714 |
lemma openin_Alexandroff_compactification: |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1715 |
"openin(Alexandroff_compactification X) V \<longleftrightarrow> |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1716 |
(\<exists>U. openin X U \<and> V = Some ` U) \<or> |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1717 |
(\<exists>C. compactin X C \<and> closedin X C \<and> V = insert None (Some ` (topspace X - C)))" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1718 |
by (auto simp: Alexandroff_compactification_def istopology_Alexandroff_open Alexandroff_open.simps) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1719 |
|
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1720 |
|
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1721 |
lemma topspace_Alexandroff_compactification [simp]: |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1722 |
"topspace(Alexandroff_compactification X) = insert None (Some ` topspace X)" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1723 |
(is "?lhs = ?rhs") |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1724 |
proof |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1725 |
show "?lhs \<subseteq> ?rhs" |
78202 | 1726 |
by (force simp: topspace_def openin_Alexandroff_compactification) |
78127
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1727 |
have "None \<in> topspace (Alexandroff_compactification X)" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1728 |
by (meson closedin_empty compactin_empty insert_subset openin_Alexandroff_compactification openin_subset) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1729 |
moreover have "Some x \<in> topspace (Alexandroff_compactification X)" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1730 |
if "x \<in> topspace X" for x |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1731 |
by (meson that imageI openin_Alexandroff_compactification openin_subset openin_topspace subsetD) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1732 |
ultimately show "?rhs \<subseteq> ?lhs" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1733 |
by (auto simp: image_subset_iff) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1734 |
qed |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1735 |
|
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1736 |
lemma closedin_Alexandroff_compactification: |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1737 |
"closedin (Alexandroff_compactification X) C \<longleftrightarrow> |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1738 |
(\<exists>K. compactin X K \<and> closedin X K \<and> C = Some ` K) \<or> |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1739 |
(\<exists>U. openin X U \<and> C = topspace(Alexandroff_compactification X) - Some ` U)" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1740 |
(is "?lhs \<longleftrightarrow> ?rhs") |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1741 |
proof |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1742 |
show "?lhs \<Longrightarrow> ?rhs" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1743 |
apply (clarsimp simp: closedin_def openin_Alexandroff_compactification) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1744 |
by (smt (verit) Diff_Diff_Int None_notin_image_Some image_set_diff inf.absorb_iff2 inj_Some insert_Diff_if subset_insert) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1745 |
show "?rhs \<Longrightarrow> ?lhs" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1746 |
using openin_subset |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1747 |
by (fastforce simp: closedin_def openin_Alexandroff_compactification) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1748 |
qed |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1749 |
|
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1750 |
lemma openin_Alexandroff_compactification_image_Some [simp]: |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1751 |
"openin(Alexandroff_compactification X) (Some ` U) \<longleftrightarrow> openin X U" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1752 |
by (auto simp: openin_Alexandroff_compactification inj_image_eq_iff) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1753 |
|
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1754 |
lemma closedin_Alexandroff_compactification_image_Some [simp]: |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1755 |
"closedin (Alexandroff_compactification X) (Some ` K) \<longleftrightarrow> compactin X K \<and> closedin X K" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1756 |
by (auto simp: closedin_Alexandroff_compactification inj_image_eq_iff) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1757 |
|
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1758 |
lemma open_map_Some: "open_map X (Alexandroff_compactification X) Some" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1759 |
using open_map_def openin_Alexandroff_compactification by blast |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1760 |
|
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1761 |
lemma continuous_map_Some: "continuous_map X (Alexandroff_compactification X) Some" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1762 |
unfolding continuous_map_def |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1763 |
proof (intro conjI strip) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1764 |
fix U |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1765 |
assume "openin (Alexandroff_compactification X) U" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1766 |
then consider V where "openin X V" "U = Some ` V" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1767 |
| C where "compactin X C" "closedin X C" "U = insert None (Some ` (topspace X - C))" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1768 |
by (auto simp: openin_Alexandroff_compactification) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1769 |
then show "openin X {x \<in> topspace X. Some x \<in> U}" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1770 |
proof cases |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1771 |
case 1 |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1772 |
then show ?thesis |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1773 |
using openin_subopen openin_subset by fastforce |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1774 |
next |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1775 |
case 2 |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1776 |
then show ?thesis |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1777 |
by (simp add: closedin_def image_iff set_diff_eq) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1778 |
qed |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1779 |
qed auto |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1780 |
|
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1781 |
|
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1782 |
lemma embedding_map_Some: "embedding_map X (Alexandroff_compactification X) Some" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1783 |
by (simp add: continuous_map_Some injective_open_imp_embedding_map open_map_Some) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1784 |
|
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1785 |
lemma compact_space_Alexandroff_compactification [simp]: |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1786 |
"compact_space(Alexandroff_compactification X)" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1787 |
proof (clarsimp simp: compact_space_alt image_subset_iff) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1788 |
fix \<U> U |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1789 |
assume ope [rule_format]: "\<forall>U\<in>\<U>. openin (Alexandroff_compactification X) U" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1790 |
and cover: "\<forall>x\<in>topspace X. \<exists>X\<in>\<U>. Some x \<in> X" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1791 |
and "U \<in> \<U>" "None \<in> U" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1792 |
then have Usub: "U \<subseteq> insert None (Some ` topspace X)" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1793 |
by (metis openin_subset topspace_Alexandroff_compactification) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1794 |
with ope [OF \<open>U \<in> \<U>\<close>] \<open>None \<in> U\<close> |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1795 |
obtain C where C: "compactin X C \<and> closedin X C \<and> |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1796 |
insert None (Some ` topspace X) - U = Some ` C" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1797 |
by (auto simp: openin_closedin closedin_Alexandroff_compactification) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1798 |
then have D: "compactin (Alexandroff_compactification X) (insert None (Some ` topspace X) - U)" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1799 |
by (metis continuous_map_Some image_compactin) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1800 |
consider V where "openin X V" "U = Some ` V" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1801 |
| C where "compactin X C" "closedin X C" "U = insert None (Some ` (topspace X - C))" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1802 |
using ope [OF \<open>U \<in> \<U>\<close>] by (auto simp: openin_Alexandroff_compactification) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1803 |
then show "\<exists>\<F>. finite \<F> \<and> \<F> \<subseteq> \<U> \<and> (\<exists>X\<in>\<F>. None \<in> X) \<and> (\<forall>x\<in>topspace X. \<exists>X\<in>\<F>. Some x \<in> X)" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1804 |
proof cases |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1805 |
case 1 |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1806 |
then show ?thesis |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1807 |
using \<open>None \<in> U\<close> by blast |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1808 |
next |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1809 |
case 2 |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1810 |
obtain \<F> where "finite \<F>" "\<F> \<subseteq> \<U>" "insert None (Some ` topspace X) - U \<subseteq> \<Union>\<F>" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1811 |
by (smt (verit, del_insts) C D Union_iff compactinD compactin_subset_topspace cover image_subset_iff ope subsetD) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1812 |
with \<open>U \<in> \<U>\<close> show ?thesis |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1813 |
by (rule_tac x="insert U \<F>" in exI) auto |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1814 |
qed |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1815 |
qed |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1816 |
|
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1817 |
lemma topspace_Alexandroff_compactification_delete: |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1818 |
"topspace(Alexandroff_compactification X) - {None} = Some ` topspace X" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1819 |
by simp |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1820 |
|
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1821 |
lemma Alexandroff_compactification_dense: |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1822 |
assumes "\<not> compact_space X" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1823 |
shows "(Alexandroff_compactification X) closure_of (Some ` topspace X) = |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1824 |
topspace(Alexandroff_compactification X)" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1825 |
unfolding topspace_Alexandroff_compactification_delete [symmetric] |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1826 |
proof (intro one_point_compactification_dense) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1827 |
show "\<not> compactin (Alexandroff_compactification X) (topspace (Alexandroff_compactification X) - {None})" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1828 |
using assms compact_space_proper_map_preimage compact_space_subtopology embedding_map_Some embedding_map_def homeomorphic_imp_proper_map by fastforce |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1829 |
qed auto |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1830 |
|
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1831 |
|
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1832 |
lemma t0_space_one_point_compactification: |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1833 |
assumes "compact_space X \<and> openin X (topspace X - {a})" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1834 |
shows "t0_space X \<longleftrightarrow> t0_space (subtopology X (topspace X - {a}))" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1835 |
(is "?lhs \<longleftrightarrow> ?rhs") |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1836 |
proof |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1837 |
show "?lhs \<Longrightarrow> ?rhs" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1838 |
using t0_space_subtopology by blast |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1839 |
show "?rhs \<Longrightarrow> ?lhs" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1840 |
using assms |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1841 |
unfolding t0_space_def by (bestsimp simp flip: Int_Diff dest: openin_trans_full) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1842 |
qed |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1843 |
|
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1844 |
lemma t0_space_Alexandroff_compactification [simp]: |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1845 |
"t0_space (Alexandroff_compactification X) \<longleftrightarrow> t0_space X" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1846 |
using t0_space_one_point_compactification [of "Alexandroff_compactification X" None] |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1847 |
using embedding_map_Some embedding_map_imp_homeomorphic_space homeomorphic_t0_space by fastforce |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1848 |
|
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1849 |
lemma t1_space_one_point_compactification: |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1850 |
assumes Xa: "openin X (topspace X - {a})" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1851 |
and \<section>: "\<And>K. \<lbrakk>compactin (subtopology X (topspace X - {a})) K; closedin (subtopology X (topspace X - {a})) K\<rbrakk> \<Longrightarrow> closedin X K" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1852 |
shows "t1_space X \<longleftrightarrow> t1_space (subtopology X (topspace X - {a}))" (is "?lhs \<longleftrightarrow> ?rhs") |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1853 |
proof |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1854 |
show "?lhs \<Longrightarrow> ?rhs" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1855 |
using t1_space_subtopology by blast |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1856 |
assume R: ?rhs |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1857 |
show ?lhs |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1858 |
unfolding t1_space_closedin_singleton |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1859 |
proof (intro strip) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1860 |
fix x |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1861 |
assume "x \<in> topspace X" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1862 |
show "closedin X {x}" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1863 |
proof (cases "x=a") |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1864 |
case True |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1865 |
then show ?thesis |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1866 |
using \<open>x \<in> topspace X\<close> Xa closedin_def by blast |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1867 |
next |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1868 |
case False |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1869 |
show ?thesis |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1870 |
by (simp add: "\<section>" False R \<open>x \<in> topspace X\<close> closedin_t1_singleton) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1871 |
qed |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1872 |
qed |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1873 |
qed |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1874 |
|
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1875 |
lemma closedin_Alexandroff_I: |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1876 |
assumes "compactin (Alexandroff_compactification X) K" "K \<subseteq> Some ` topspace X" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1877 |
"closedin (Alexandroff_compactification X) T" "K = T \<inter> Some ` topspace X" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1878 |
shows "closedin (Alexandroff_compactification X) K" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1879 |
proof - |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1880 |
obtain S where S: "S \<subseteq> topspace X" "K = Some ` S" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1881 |
by (meson \<open>K \<subseteq> Some ` topspace X\<close> subset_imageE) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1882 |
with assms have "compactin X S" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1883 |
by (metis compactin_subtopology embedding_map_Some embedding_map_def homeomorphic_map_compactness) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1884 |
moreover have "closedin X S" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1885 |
using assms S |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1886 |
by (metis closedin_subtopology embedding_map_Some embedding_map_def homeomorphic_map_closedness) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1887 |
ultimately show ?thesis |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1888 |
by (simp add: S) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1889 |
qed |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1890 |
|
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1891 |
|
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1892 |
lemma t1_space_Alexandroff_compactification [simp]: |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1893 |
"t1_space(Alexandroff_compactification X) \<longleftrightarrow> t1_space X" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1894 |
proof - |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1895 |
have "openin (Alexandroff_compactification X) (topspace (Alexandroff_compactification X) - {None})" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1896 |
by auto |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1897 |
then show ?thesis |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1898 |
using t1_space_one_point_compactification [of "Alexandroff_compactification X" None] |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1899 |
using embedding_map_Some embedding_map_imp_homeomorphic_space homeomorphic_t1_space |
78202 | 1900 |
by (fastforce simp: compactin_subtopology closedin_Alexandroff_I closedin_subtopology) |
78127
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1901 |
qed |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1902 |
|
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1903 |
|
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1904 |
lemma kc_space_one_point_compactification: |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1905 |
assumes "compact_space X" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1906 |
and ope: "openin X (topspace X - {a})" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1907 |
and \<section>: "\<And>K. \<lbrakk>compactin (subtopology X (topspace X - {a})) K; closedin (subtopology X (topspace X - {a})) K\<rbrakk> |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1908 |
\<Longrightarrow> closedin X K" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1909 |
shows "kc_space X \<longleftrightarrow> |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1910 |
k_space (subtopology X (topspace X - {a})) \<and> kc_space (subtopology X (topspace X - {a}))" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1911 |
proof - |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1912 |
have "closedin X K" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1913 |
if "kc_space (subtopology X (topspace X - {a}))" and "compactin X K" "a \<notin> K" for K |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1914 |
using that unfolding kc_space_def |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1915 |
by (metis "\<section>" Diff_empty compactin_subspace compactin_subtopology subset_Diff_insert) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1916 |
then show ?thesis |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1917 |
by (metis \<open>compact_space X\<close> kc_space_one_point_compactification_gen ope) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1918 |
qed |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1919 |
|
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1920 |
lemma kc_space_Alexandroff_compactification: |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1921 |
"kc_space(Alexandroff_compactification X) \<longleftrightarrow> (k_space X \<and> kc_space X)" (is "kc_space ?Y = _") |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1922 |
proof - |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1923 |
have "kc_space (Alexandroff_compactification X) \<longleftrightarrow> |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1924 |
(k_space (subtopology ?Y (topspace ?Y - {None})) \<and> kc_space (subtopology ?Y (topspace ?Y - {None})))" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1925 |
by (rule kc_space_one_point_compactification) (auto simp: compactin_subtopology closedin_subtopology closedin_Alexandroff_I) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1926 |
also have "... \<longleftrightarrow> k_space X \<and> kc_space X" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1927 |
using embedding_map_Some embedding_map_imp_homeomorphic_space homeomorphic_k_space homeomorphic_kc_space by simp blast |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1928 |
finally show ?thesis . |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1929 |
qed |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1930 |
|
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1931 |
|
78274
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
1932 |
proposition regular_space_one_point_compactification: |
78127
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1933 |
assumes "compact_space X" and ope: "openin X (topspace X - {a})" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1934 |
and \<section>: "\<And>K. \<lbrakk>compactin (subtopology X (topspace X - {a})) K; closedin (subtopology X (topspace X - {a})) K\<rbrakk> \<Longrightarrow> closedin X K" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1935 |
shows "regular_space X \<longleftrightarrow> |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1936 |
regular_space (subtopology X (topspace X - {a})) \<and> locally_compact_space (subtopology X (topspace X - {a}))" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1937 |
(is "?lhs \<longleftrightarrow> ?rhs") |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1938 |
proof |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1939 |
show "?lhs \<Longrightarrow> ?rhs" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1940 |
using assms(1) compact_imp_locally_compact_space locally_compact_space_open_subset ope regular_space_subtopology by blast |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1941 |
assume R: ?rhs |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1942 |
let ?Xa = "subtopology X (topspace X - {a})" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1943 |
show ?lhs |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1944 |
unfolding neighbourhood_base_of_closedin [symmetric] neighbourhood_base_of imp_conjL |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1945 |
proof (intro strip) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1946 |
fix W x |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1947 |
assume "openin X W" and "x \<in> W" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1948 |
show "\<exists>U V. openin X U \<and> closedin X V \<and> x \<in> U \<and> U \<subseteq> V \<and> V \<subseteq> W" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1949 |
proof (cases "x=a") |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1950 |
case True |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1951 |
have "compactin ?Xa (topspace X - W)" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1952 |
using \<open>openin X W\<close> assms(1) closedin_compact_space |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1953 |
by (metis Diff_mono True \<open>x \<in> W\<close> compactin_subtopology empty_subsetI insert_subset openin_closedin_eq order_refl) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1954 |
then obtain V K where V: "openin ?Xa V" and K: "compactin ?Xa K" "closedin ?Xa K" and "topspace X - W \<subseteq> V" "V \<subseteq> K" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1955 |
by (metis locally_compact_space_compact_closed_compact R) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1956 |
show ?thesis |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1957 |
proof (intro exI conjI) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1958 |
show "openin X (topspace X - K)" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1959 |
using "\<section>" K by blast |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1960 |
show "closedin X (topspace X - V)" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1961 |
using V ope openin_trans_full by blast |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1962 |
show "x \<in> topspace X - K" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1963 |
proof (rule) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1964 |
show "x \<in> topspace X" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1965 |
using \<open>openin X W\<close> \<open>x \<in> W\<close> openin_subset by blast |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1966 |
show "x \<notin> K" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1967 |
using K True closedin_imp_subset by blast |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1968 |
qed |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1969 |
show "topspace X - K \<subseteq> topspace X - V" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1970 |
by (simp add: Diff_mono \<open>V \<subseteq> K\<close>) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1971 |
show "topspace X - V \<subseteq> W" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1972 |
using \<open>topspace X - W \<subseteq> V\<close> by auto |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1973 |
qed |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1974 |
next |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1975 |
case False |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1976 |
have "openin ?Xa ((topspace X - {a}) \<inter> W)" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1977 |
using \<open>openin X W\<close> openin_subtopology_Int2 by blast |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1978 |
moreover have "x \<in> (topspace X - {a}) \<inter> W" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1979 |
using \<open>openin X W\<close> \<open>x \<in> W\<close> openin_subset False by blast |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1980 |
ultimately obtain U V where "openin ?Xa U" "compactin ?Xa V" "closedin ?Xa V" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1981 |
"x \<in> U" "U \<subseteq> V" "V \<subseteq> (topspace X - {a}) \<inter> W" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1982 |
using R locally_compact_regular_space_neighbourhood_base neighbourhood_base_of |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1983 |
by (metis (no_types, lifting)) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1984 |
then show ?thesis |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1985 |
by (meson "\<section>" le_infE ope openin_trans_full) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1986 |
qed |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1987 |
qed |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1988 |
qed |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1989 |
|
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1990 |
|
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1991 |
lemma regular_space_Alexandroff_compactification: |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1992 |
"regular_space(Alexandroff_compactification X) \<longleftrightarrow> regular_space X \<and> locally_compact_space X" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1993 |
(is "regular_space ?Y = ?rhs") |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1994 |
proof - |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1995 |
have "regular_space ?Y \<longleftrightarrow> |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1996 |
regular_space (subtopology ?Y (topspace ?Y - {None})) \<and> locally_compact_space (subtopology ?Y (topspace ?Y - {None}))" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1997 |
by (rule regular_space_one_point_compactification) (auto simp: compactin_subtopology closedin_subtopology closedin_Alexandroff_I) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1998 |
also have "... \<longleftrightarrow> regular_space X \<and> locally_compact_space X" |
78200
264f2b69d09c
New and generalised analysis lemmas
paulson <lp15@cam.ac.uk>
parents:
78128
diff
changeset
|
1999 |
by (metis embedding_map_Some embedding_map_imp_homeomorphic_space homeomorphic_locally_compact_space |
264f2b69d09c
New and generalised analysis lemmas
paulson <lp15@cam.ac.uk>
parents:
78128
diff
changeset
|
2000 |
homeomorphic_regular_space topspace_Alexandroff_compactification_delete) |
78127
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2001 |
finally show ?thesis . |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2002 |
qed |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2003 |
|
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2004 |
|
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2005 |
lemma Hausdorff_space_one_point_compactification: |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2006 |
assumes "compact_space X" and "openin X (topspace X - {a})" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2007 |
and "\<And>K. \<lbrakk>compactin (subtopology X (topspace X - {a})) K; closedin (subtopology X (topspace X - {a})) K\<rbrakk> \<Longrightarrow> closedin X K" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2008 |
shows "Hausdorff_space X \<longleftrightarrow> |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2009 |
Hausdorff_space (subtopology X (topspace X - {a})) \<and> locally_compact_space (subtopology X (topspace X - {a}))" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2010 |
(is "?lhs \<longleftrightarrow> ?rhs") |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2011 |
proof |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2012 |
show ?rhs if ?lhs |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2013 |
proof - |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2014 |
have "locally_compact_space (subtopology X (topspace X - {a}))" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2015 |
using assms that compact_imp_locally_compact_space locally_compact_space_open_subset |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2016 |
by blast |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2017 |
with that show ?rhs |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2018 |
by (simp add: Hausdorff_space_subtopology) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2019 |
qed |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2020 |
next |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2021 |
show "?rhs \<Longrightarrow> ?lhs" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2022 |
by (metis assms locally_compact_Hausdorff_or_regular regular_space_one_point_compactification |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2023 |
regular_t1_eq_Hausdorff_space t1_space_one_point_compactification) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2024 |
qed |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2025 |
|
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2026 |
lemma Hausdorff_space_Alexandroff_compactification: |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2027 |
"Hausdorff_space(Alexandroff_compactification X) \<longleftrightarrow> Hausdorff_space X \<and> locally_compact_space X" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2028 |
by (meson compact_Hausdorff_imp_regular_space compact_space_Alexandroff_compactification |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2029 |
locally_compact_Hausdorff_or_regular regular_space_Alexandroff_compactification |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2030 |
regular_t1_eq_Hausdorff_space t1_space_Alexandroff_compactification) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2031 |
|
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2032 |
lemma completely_regular_space_Alexandroff_compactification: |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2033 |
"completely_regular_space(Alexandroff_compactification X) \<longleftrightarrow> |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2034 |
completely_regular_space X \<and> locally_compact_space X" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2035 |
by (metis regular_space_Alexandroff_compactification completely_regular_eq_regular_space |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2036 |
compact_imp_locally_compact_space compact_space_Alexandroff_compactification) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2037 |
|
78274
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
2038 |
proposition Hausdorff_space_one_point_compactification_asymmetric_prod: |
78127
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2039 |
assumes "compact_space X" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2040 |
shows "Hausdorff_space X \<longleftrightarrow> |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2041 |
kc_space (prod_topology X (subtopology X (topspace X - {a}))) \<and> |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2042 |
k_space (prod_topology X (subtopology X (topspace X - {a})))" (is "?lhs \<longleftrightarrow> ?rhs") |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2043 |
proof (cases "a \<in> topspace X") |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2044 |
case True |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2045 |
show ?thesis |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2046 |
proof |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2047 |
show ?rhs if ?lhs |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2048 |
proof |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2049 |
show "kc_space (prod_topology X (subtopology X (topspace X - {a})))" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2050 |
using Hausdorff_imp_kc_space kc_space_prod_topology_right kc_space_subtopology that by blast |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2051 |
show "k_space (prod_topology X (subtopology X (topspace X - {a})))" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2052 |
by (meson Hausdorff_imp_kc_space assms compact_imp_locally_compact_space k_space_prod_topology_left |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2053 |
kc_space_one_point_compactification_gen that) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2054 |
qed |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2055 |
next |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2056 |
assume R: ?rhs |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2057 |
define D where "D \<equiv> (\<lambda>x. (x,x)) ` (topspace X - {a})" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2058 |
show ?lhs |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2059 |
proof (cases "topspace X = {a}") |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2060 |
case True |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2061 |
then show ?thesis |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2062 |
by (simp add: Hausdorff_space_def) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2063 |
next |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2064 |
case False |
78336 | 2065 |
with R True have "kc_space X" |
78127
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2066 |
using kc_space_retraction_map_image [of "prod_topology X (subtopology X (topspace X - {a}))" X fst] |
78336 | 2067 |
by (metis Diff_subset insert_Diff retraction_map_fst topspace_discrete_topology topspace_subtopology_subset) |
78127
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2068 |
have "closedin (subtopology (prod_topology X (subtopology X (topspace X - {a}))) K) (K \<inter> D)" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2069 |
if "compactin (prod_topology X (subtopology X (topspace X - {a}))) K" for K |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2070 |
proof (intro closedin_subtopology_Int_subset[where V=K] closedin_subset_topspace) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2071 |
show "fst ` K \<times> snd ` K \<inter> D \<subseteq> fst ` K \<times> snd ` K" "K \<subseteq> fst ` K \<times> snd ` K" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2072 |
by force+ |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2073 |
have eq: "(fst ` K \<times> snd ` K \<inter> D) = ((\<lambda>x. (x,x)) ` (fst ` K \<inter> snd ` K))" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2074 |
using compactin_subset_topspace that by (force simp: D_def image_iff) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2075 |
have "compactin (prod_topology X (subtopology X (topspace X - {a}))) (fst ` K \<times> snd ` K \<inter> D)" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2076 |
unfolding eq |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2077 |
proof (rule image_compactin [of "subtopology X (topspace X - {a})"]) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2078 |
have "compactin X (fst ` K)" "compactin X (snd ` K)" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2079 |
by (meson compactin_subtopology continuous_map_fst continuous_map_snd image_compactin that)+ |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2080 |
moreover have "fst ` K \<inter> snd ` K \<subseteq> topspace X - {a}" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2081 |
using compactin_subset_topspace that by force |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2082 |
ultimately |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2083 |
show "compactin (subtopology X (topspace X - {a})) (fst ` K \<inter> snd ` K)" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2084 |
unfolding compactin_subtopology |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2085 |
by (meson \<open>kc_space X\<close> closed_Int_compactin kc_space_def) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2086 |
show "continuous_map (subtopology X (topspace X - {a})) (prod_topology X (subtopology X (topspace X - {a}))) (\<lambda>x. (x,x))" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2087 |
by (simp add: continuous_map_paired) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2088 |
qed |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2089 |
then show "closedin (prod_topology X (subtopology X (topspace X - {a}))) (fst ` K \<times> snd ` K \<inter> D)" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2090 |
using R compactin_imp_closedin_gen by blast |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2091 |
qed |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2092 |
moreover have "D \<subseteq> topspace X \<times> (topspace X \<inter> (topspace X - {a}))" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2093 |
by (auto simp: D_def) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2094 |
ultimately have *: "closedin (prod_topology X (subtopology X (topspace X - {a}))) D" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2095 |
using R by (auto simp: k_space) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2096 |
have "x=y" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2097 |
if "x \<in> topspace X" "y \<in> topspace X" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2098 |
and \<section>: "\<And>T. \<lbrakk>(x,y) \<in> T; openin (prod_topology X X) T\<rbrakk> \<Longrightarrow> \<exists>z \<in> topspace X. (z,z) \<in> T" for x y |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2099 |
proof (cases "x=a \<and> y=a") |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2100 |
case False |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2101 |
then consider "x\<noteq>a" | "y\<noteq>a" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2102 |
by blast |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2103 |
then show ?thesis |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2104 |
proof cases |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2105 |
case 1 |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2106 |
have "\<exists>z \<in> topspace X - {a}. (z,z) \<in> T" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2107 |
if "(y,x) \<in> T" "openin (prod_topology X (subtopology X (topspace X - {a}))) T" for T |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2108 |
proof - |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2109 |
have "(x,y) \<in> {z \<in> topspace (prod_topology X X). (snd z,fst z) \<in> T \<inter> topspace X \<times> (topspace X - {a})}" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2110 |
by (simp add: "1" \<open>x \<in> topspace X\<close> \<open>y \<in> topspace X\<close> that) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2111 |
moreover have "openin (prod_topology X X) {z \<in> topspace (prod_topology X X). (snd z,fst z) \<in> T \<inter> topspace X \<times> (topspace X - {a})}" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2112 |
proof (rule openin_continuous_map_preimage) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2113 |
show "continuous_map (prod_topology X X) (prod_topology X X) (\<lambda>x. (snd x, fst x))" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2114 |
by (simp add: continuous_map_fst continuous_map_pairedI continuous_map_snd) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2115 |
have "openin (prod_topology X X) (topspace X \<times> (topspace X - {a}))" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2116 |
using \<open>kc_space X\<close> assms kc_space_one_point_compactification_gen openin_prod_Times_iff by fastforce |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2117 |
moreover have "openin (prod_topology X X) T" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2118 |
using kc_space_one_point_compactification_gen [OF \<open>compact_space X\<close>] \<open>kc_space X\<close> that |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2119 |
by (metis openin_prod_Times_iff openin_topspace openin_trans_full prod_topology_subtopology(2)) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2120 |
ultimately show "openin (prod_topology X X) (T \<inter> topspace X \<times> (topspace X - {a}))" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2121 |
by blast |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2122 |
qed |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2123 |
ultimately show ?thesis |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2124 |
by (smt (verit) \<section> Int_iff fst_conv mem_Collect_eq mem_Sigma_iff snd_conv) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2125 |
qed |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2126 |
then have "(y,x) \<in> prod_topology X (subtopology X (topspace X - {a})) closure_of D" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2127 |
by (simp add: "1" D_def in_closure_of that) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2128 |
then show ?thesis |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2129 |
using that "*" D_def closure_of_closedin by fastforce |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2130 |
next |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2131 |
case 2 |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2132 |
have "\<exists>z \<in> topspace X - {a}. (z,z) \<in> T" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2133 |
if "(x,y) \<in> T" "openin (prod_topology X (subtopology X (topspace X - {a}))) T" for T |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2134 |
proof - |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2135 |
have "openin (prod_topology X X) (topspace X \<times> (topspace X - {a}))" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2136 |
using \<open>kc_space X\<close> assms kc_space_one_point_compactification_gen openin_prod_Times_iff by fastforce |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2137 |
moreover have XXT: "openin (prod_topology X X) T" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2138 |
using kc_space_one_point_compactification_gen [OF \<open>compact_space X\<close>] \<open>kc_space X\<close> that |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2139 |
by (metis openin_prod_Times_iff openin_topspace openin_trans_full prod_topology_subtopology(2)) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2140 |
ultimately have "openin (prod_topology X X) (T \<inter> topspace X \<times> (topspace X - {a}))" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2141 |
by blast |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2142 |
then show ?thesis |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2143 |
by (smt (verit) "\<section>" Diff_subset XXT mem_Sigma_iff openin_subset subsetD that topspace_prod_topology topspace_subtopology_subset) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2144 |
qed |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2145 |
then have "(x,y) \<in> prod_topology X (subtopology X (topspace X - {a})) closure_of D" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2146 |
by (simp add: "2" D_def in_closure_of that) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2147 |
then show ?thesis |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2148 |
using that "*" D_def closure_of_closedin by fastforce |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2149 |
qed |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2150 |
qed auto |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2151 |
then show ?thesis |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2152 |
unfolding Hausdorff_space_closedin_diagonal closure_of_subset_eq [symmetric] |
78202 | 2153 |
by (force simp: closure_of_def) |
78127
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2154 |
qed |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2155 |
qed |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2156 |
next |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2157 |
case False |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2158 |
then show ?thesis |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2159 |
by (simp add: assms compact_imp_k_space compact_space_prod_topology kc_space_compact_prod_topology) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2160 |
qed |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2161 |
|
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2162 |
|
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2163 |
lemma Hausdorff_space_Alexandroff_compactification_asymmetric_prod: |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2164 |
"Hausdorff_space(Alexandroff_compactification X) \<longleftrightarrow> |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2165 |
kc_space(prod_topology (Alexandroff_compactification X) X) \<and> |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2166 |
k_space(prod_topology (Alexandroff_compactification X) X)" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2167 |
(is "Hausdorff_space ?Y = ?rhs") |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2168 |
proof - |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2169 |
have *: "subtopology (Alexandroff_compactification X) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2170 |
(topspace (Alexandroff_compactification X) - |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2171 |
{None}) homeomorphic_space X" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2172 |
using embedding_map_Some embedding_map_imp_homeomorphic_space homeomorphic_space_sym by fastforce |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2173 |
have "Hausdorff_space (Alexandroff_compactification X) \<longleftrightarrow> |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2174 |
(kc_space (prod_topology ?Y (subtopology ?Y (topspace ?Y - {None}))) \<and> |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2175 |
k_space (prod_topology ?Y (subtopology ?Y (topspace ?Y - {None}))))" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2176 |
by (rule Hausdorff_space_one_point_compactification_asymmetric_prod) (auto simp: compactin_subtopology closedin_subtopology closedin_Alexandroff_I) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2177 |
also have "... \<longleftrightarrow> ?rhs" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2178 |
using homeomorphic_k_space homeomorphic_kc_space homeomorphic_space_prod_topology |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2179 |
homeomorphic_space_refl * by blast |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2180 |
finally show ?thesis . |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2181 |
qed |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2182 |
|
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2183 |
lemma kc_space_as_compactification_unique: |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2184 |
assumes "kc_space X" "compact_space X" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2185 |
shows "openin X U \<longleftrightarrow> |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2186 |
(if a \<in> U then U \<subseteq> topspace X \<and> compactin X (topspace X - U) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2187 |
else openin (subtopology X (topspace X - {a})) U)" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2188 |
proof (cases "a \<in> U") |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2189 |
case True |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2190 |
then show ?thesis |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2191 |
by (meson assms closedin_compact_space compactin_imp_closedin_gen openin_closedin_eq) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2192 |
next |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2193 |
case False |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2194 |
then show ?thesis |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2195 |
by (metis Diff_empty kc_space_one_point_compactification_gen openin_open_subtopology openin_subset subset_Diff_insert assms) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2196 |
qed |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2197 |
|
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2198 |
lemma kc_space_as_compactification_unique_explicit: |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2199 |
assumes "kc_space X" "compact_space X" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2200 |
shows "openin X U \<longleftrightarrow> |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2201 |
(if a \<in> U then U \<subseteq> topspace X \<and> |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2202 |
compactin (subtopology X (topspace X - {a})) (topspace X - U) \<and> |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2203 |
closedin (subtopology X (topspace X - {a})) (topspace X - U) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2204 |
else openin (subtopology X (topspace X - {a})) U)" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2205 |
apply (simp add: kc_space_subtopology compactin_imp_closedin_gen assms compactin_subtopology cong: conj_cong) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2206 |
by (metis Diff_mono assms bot.extremum insert_subset kc_space_as_compactification_unique subset_refl) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2207 |
|
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2208 |
lemma Alexandroff_compactification_unique: |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2209 |
assumes "kc_space X" "compact_space X" and a: "a \<in> topspace X" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2210 |
shows "Alexandroff_compactification (subtopology X (topspace X - {a})) homeomorphic_space X" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2211 |
(is "?Y homeomorphic_space X") |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2212 |
proof - |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2213 |
have [simp]: "topspace X \<inter> (topspace X - {a}) = topspace X - {a}" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2214 |
by auto |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2215 |
have [simp]: "insert None (Some ` A) = insert None (Some ` B) \<longleftrightarrow> A = B" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2216 |
"insert None (Some ` A) \<noteq> Some ` B" for A B |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2217 |
by auto |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2218 |
have "quotient_map X ?Y (\<lambda>x. if x = a then None else Some x)" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2219 |
unfolding quotient_map_def |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2220 |
proof (intro conjI strip) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2221 |
show "(\<lambda>x. if x = a then None else Some x) ` topspace X = topspace ?Y" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2222 |
using \<open>a \<in> topspace X\<close> by force |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2223 |
show "openin X {x \<in> topspace X. (if x = a then None else Some x) \<in> U} = openin ?Y U" (is "?L = ?R") |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2224 |
if "U \<subseteq> topspace ?Y" for U |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2225 |
proof (cases "None \<in> U") |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2226 |
case True |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2227 |
then obtain T where T[simp]: "U = insert None (Some ` T)" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2228 |
by (metis Int_insert_right UNIV_I UNIV_option_conv inf.orderE inf_le2 subsetI subset_imageE) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2229 |
have Tsub: "T \<subseteq> topspace X - {a}" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2230 |
using in_these_eq that by auto |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2231 |
then have "{x \<in> topspace X. (if x = a then None else Some x) \<in> U} = insert a T" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2232 |
by (auto simp: a image_iff cong: conj_cong) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2233 |
then have "?L \<longleftrightarrow> openin X (insert a T)" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2234 |
by metis |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2235 |
also have "\<dots> \<longleftrightarrow> ?R" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2236 |
using Tsub assms |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2237 |
apply (simp add: openin_Alexandroff_compactification kc_space_as_compactification_unique_explicit [where a=a]) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2238 |
by (smt (verit, best) Diff_insert2 Diff_subset closedin_imp_subset double_diff) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2239 |
finally show ?thesis . |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2240 |
next |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2241 |
case False |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2242 |
then obtain T where [simp]: "U = Some ` T" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2243 |
by (metis Int_insert_right UNIV_I UNIV_option_conv inf.orderE inf_le2 subsetI subset_imageE) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2244 |
have **: "\<And>V. openin X V \<Longrightarrow> openin X (V - {a})" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2245 |
by (simp add: assms compactin_imp_closedin_gen openin_diff) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2246 |
have Tsub: "T \<subseteq> topspace X - {a}" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2247 |
using in_these_eq that by auto |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2248 |
then have "{x \<in> topspace X. (if x = a then None else Some x) \<in> U} = T" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2249 |
by (auto simp: image_iff cong: conj_cong) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2250 |
then show ?thesis |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2251 |
by (simp add: "**" Tsub openin_open_subtopology) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2252 |
qed |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2253 |
qed |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2254 |
moreover have "inj_on (\<lambda>x. if x = a then None else Some x) (topspace X)" |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2255 |
by (auto simp: inj_on_def) |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2256 |
ultimately show ?thesis |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2257 |
using homeomorphic_space_sym homeomorphic_space homeomorphic_map_def by blast |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2258 |
qed |
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2259 |
|
78202 | 2260 |
subsection\<open>Extending continuous maps "pointwise" in a regular space\<close> |
2261 |
||
2262 |
lemma continuous_map_on_intermediate_closure_of: |
|
2263 |
assumes Y: "regular_space Y" |
|
2264 |
and T: "T \<subseteq> X closure_of S" |
|
2265 |
and f: "\<And>t. t \<in> T \<Longrightarrow> limitin Y f (f t) (atin_within X t S)" |
|
2266 |
shows "continuous_map (subtopology X T) Y f" |
|
2267 |
proof (clarsimp simp add: continuous_map_atin) |
|
2268 |
fix a |
|
2269 |
assume "a \<in> topspace X" and "a \<in> T" |
|
2270 |
have "f ` T \<subseteq> topspace Y" |
|
2271 |
by (metis f image_subsetI limitin_topspace) |
|
2272 |
have "\<forall>\<^sub>F x in atin_within X a T. f x \<in> W" |
|
2273 |
if W: "openin Y W" "f a \<in> W" for W |
|
2274 |
proof - |
|
2275 |
obtain V C where "openin Y V" "closedin Y C" "f a \<in> V" "V \<subseteq> C" "C \<subseteq> W" |
|
2276 |
by (metis Y W neighbourhood_base_of neighbourhood_base_of_closedin) |
|
2277 |
have "\<forall>\<^sub>F x in atin_within X a S. f x \<in> V" |
|
2278 |
by (metis \<open>a \<in> T\<close> \<open>f a \<in> V\<close> \<open>openin Y V\<close> f limitin_def) |
|
2279 |
then obtain U where "openin X U" "a \<in> U" and U: "\<forall>x \<in> U - {a}. x \<in> S \<longrightarrow> f x \<in> V" |
|
2280 |
by (smt (verit) Diff_iff \<open>a \<in> topspace X\<close> eventually_atin_within insert_iff) |
|
2281 |
moreover have "f z \<in> W" if "z \<in> U" "z \<noteq> a" "z \<in> T" for z |
|
2282 |
proof - |
|
2283 |
have "z \<in> topspace X" |
|
2284 |
using \<open>openin X U\<close> openin_subset \<open>z \<in> U\<close> by blast |
|
2285 |
then have "f z \<in> topspace Y" |
|
2286 |
using \<open>f ` T \<subseteq> topspace Y\<close> \<open>z \<in> T\<close> by blast |
|
2287 |
{ assume "f z \<in> topspace Y" "f z \<notin> C" |
|
2288 |
then have "\<forall>\<^sub>F x in atin_within X z S. f x \<in> topspace Y - C" |
|
2289 |
by (metis Diff_iff \<open>closedin Y C\<close> closedin_def f limitinD \<open>z \<in> T\<close>) |
|
2290 |
then obtain U' where U': "openin X U'" "z \<in> U'" |
|
2291 |
"\<And>x. x \<in> U' - {z} \<Longrightarrow> x \<in> S \<Longrightarrow> f x \<notin> C" |
|
2292 |
by (smt (verit) Diff_iff \<open>z \<in> topspace X\<close> eventually_atin_within insertCI) |
|
2293 |
then have *: "\<And>D. z \<in> D \<and> openin X D \<Longrightarrow> \<exists>y. y \<in> S \<and> y \<in> D" |
|
2294 |
by (meson T in_closure_of subsetD \<open>z \<in> T\<close>) |
|
2295 |
have False |
|
2296 |
using * [of "U \<inter> U'"] U' U \<open>V \<subseteq> C\<close> \<open>f a \<in> V\<close> \<open>f z \<notin> C\<close> \<open>openin X U\<close> that |
|
2297 |
by blast |
|
2298 |
} |
|
2299 |
then show ?thesis |
|
2300 |
using \<open>C \<subseteq> W\<close> \<open>f z \<in> topspace Y\<close> by auto |
|
2301 |
qed |
|
2302 |
ultimately have "\<exists>U. openin X U \<and> a \<in> U \<and> (\<forall>x \<in> U - {a}. x \<in> T \<longrightarrow> f x \<in> W)" |
|
2303 |
by blast |
|
2304 |
then show ?thesis |
|
2305 |
using eventually_atin_within by fastforce |
|
2306 |
qed |
|
2307 |
then show "limitin Y f (f a) (atin (subtopology X T) a)" |
|
2308 |
by (metis \<open>a \<in> T\<close> atin_subtopology_within f limitin_def) |
|
2309 |
qed |
|
2310 |
||
2311 |
||
2312 |
lemma continuous_map_on_intermediate_closure_of_eq: |
|
2313 |
assumes "regular_space Y" "S \<subseteq> T" and Tsub: "T \<subseteq> X closure_of S" |
|
2314 |
shows "continuous_map (subtopology X T) Y f \<longleftrightarrow> (\<forall>t \<in> T. limitin Y f (f t) (atin_within X t S))" |
|
2315 |
(is "?lhs \<longleftrightarrow> ?rhs") |
|
2316 |
proof |
|
2317 |
assume L: ?lhs |
|
2318 |
show ?rhs |
|
2319 |
proof (clarsimp simp add: continuous_map_atin) |
|
2320 |
fix x |
|
2321 |
assume "x \<in> T" |
|
2322 |
with L Tsub closure_of_subset_topspace |
|
2323 |
have "limitin Y f (f x) (atin (subtopology X T) x)" |
|
2324 |
by (fastforce simp: continuous_map_atin) |
|
2325 |
then show "limitin Y f (f x) (atin_within X x S)" |
|
2326 |
using \<open>x \<in> T\<close> \<open>S \<subseteq> T\<close> |
|
2327 |
by (force simp: limitin_def atin_subtopology_within eventually_atin_within) |
|
2328 |
qed |
|
2329 |
next |
|
2330 |
show "?rhs \<Longrightarrow> ?lhs" |
|
2331 |
using assms continuous_map_on_intermediate_closure_of by blast |
|
2332 |
qed |
|
2333 |
||
2334 |
lemma continuous_map_extension_pointwise_alt: |
|
2335 |
assumes \<section>: "regular_space Y" "S \<subseteq> T" "T \<subseteq> X closure_of S" |
|
2336 |
and f: "continuous_map (subtopology X S) Y f" |
|
2337 |
and lim: "\<And>t. t \<in> T-S \<Longrightarrow> \<exists>l. limitin Y f l (atin_within X t S)" |
|
2338 |
obtains g where "continuous_map (subtopology X T) Y g" "\<And>x. x \<in> S \<Longrightarrow> g x = f x" |
|
2339 |
proof - |
|
2340 |
obtain g where g: "\<And>t. t \<in> T \<and> t \<notin> S \<Longrightarrow> limitin Y f (g t) (atin_within X t S)" |
|
2341 |
by (metis Diff_iff lim) |
|
2342 |
let ?h = "\<lambda>x. if x \<in> S then f x else g x" |
|
2343 |
show thesis |
|
2344 |
proof |
|
2345 |
have T: "T \<subseteq> topspace X" |
|
2346 |
using \<section> closure_of_subset_topspace by fastforce |
|
2347 |
have "limitin Y ?h (f t) (atin_within X t S)" if "t \<in> T" "t \<in> S" for t |
|
2348 |
proof - |
|
2349 |
have "limitin Y f (f t) (atin_within X t S)" |
|
2350 |
by (meson T f limit_continuous_map_within subset_eq that) |
|
2351 |
then show ?thesis |
|
2352 |
by (simp add: eventually_atin_within limitin_def) |
|
2353 |
qed |
|
2354 |
moreover have "limitin Y ?h (g t) (atin_within X t S)" if "t \<in> T" "t \<notin> S" for t |
|
2355 |
by (smt (verit, del_insts) eventually_atin_within g limitin_def that) |
|
2356 |
ultimately show "continuous_map (subtopology X T) Y ?h" |
|
2357 |
unfolding continuous_map_on_intermediate_closure_of_eq [OF \<section>] |
|
2358 |
by (auto simp: \<section> atin_subtopology_within) |
|
2359 |
qed auto |
|
2360 |
qed |
|
2361 |
||
2362 |
||
2363 |
lemma continuous_map_extension_pointwise: |
|
2364 |
assumes "regular_space Y" "S \<subseteq> T" and Tsub: "T \<subseteq> X closure_of S" |
|
2365 |
and ex: " \<And>x. x \<in> T \<Longrightarrow> \<exists>g. continuous_map (subtopology X (insert x S)) Y g \<and> |
|
2366 |
(\<forall>x \<in> S. g x = f x)" |
|
2367 |
obtains g where "continuous_map (subtopology X T) Y g" "\<And>x. x \<in> S \<Longrightarrow> g x = f x" |
|
2368 |
proof (rule continuous_map_extension_pointwise_alt) |
|
2369 |
show "continuous_map (subtopology X S) Y f" |
|
2370 |
proof (clarsimp simp add: continuous_map_atin) |
|
2371 |
fix t |
|
2372 |
assume "t \<in> topspace X" and "t \<in> S" |
|
2373 |
then obtain g where g: "limitin Y g (g t) (atin (subtopology X (insert t S)) t)" and gf: "\<forall>x \<in> S. g x = f x" |
|
2374 |
by (metis Int_iff \<open>S \<subseteq> T\<close> continuous_map_atin ex inf.orderE insert_absorb topspace_subtopology) |
|
2375 |
with \<open>t \<in> S\<close> show "limitin Y f (f t) (atin (subtopology X S) t)" |
|
2376 |
by (simp add: limitin_def atin_subtopology_within_if eventually_atin_within gf insert_absorb) |
|
2377 |
qed |
|
2378 |
show "\<exists>l. limitin Y f l (atin_within X t S)" if "t \<in> T - S" for t |
|
2379 |
proof - |
|
2380 |
obtain g where g: "continuous_map (subtopology X (insert t S)) Y g" and gf: "\<forall>x \<in> S. g x = f x" |
|
2381 |
using \<open>S \<subseteq> T\<close> ex \<open>t \<in> T - S\<close> by force |
|
2382 |
then have "limitin Y g (g t) (atin_within X t (insert t S))" |
|
2383 |
using Tsub in_closure_of limit_continuous_map_within that by fastforce |
|
2384 |
then show ?thesis |
|
2385 |
unfolding limitin_def |
|
2386 |
by (smt (verit) eventually_atin_within gf subsetD subset_insertI) |
|
2387 |
qed |
|
2388 |
qed (use assms in auto) |
|
2389 |
||
2390 |
||
2391 |
subsection\<open>Extending Cauchy continuous functions to the closure\<close> |
|
2392 |
||
2393 |
lemma Cauchy_continuous_map_extends_to_continuous_closure_of_aux: |
|
2394 |
assumes m2: "mcomplete_of m2" and f: "Cauchy_continuous_map (submetric m1 S) m2 f" |
|
2395 |
and "S \<subseteq> mspace m1" |
|
2396 |
obtains g |
|
2397 |
where "continuous_map (subtopology (mtopology_of m1) (mtopology_of m1 closure_of S)) |
|
2398 |
(mtopology_of m2) g" "\<And>x. x \<in> S \<Longrightarrow> g x = f x" |
|
2399 |
proof (rule continuous_map_extension_pointwise_alt) |
|
2400 |
interpret L: Metric_space12 "mspace m1" "mdist m1" "mspace m2" "mdist m2" |
|
2401 |
by (simp add: Metric_space12_mspace_mdist) |
|
2402 |
interpret S: Metric_space "S \<inter> mspace m1" "mdist m1" |
|
2403 |
by (simp add: L.M1.subspace) |
|
2404 |
show "regular_space (mtopology_of m2)" |
|
2405 |
by (simp add: Metric_space.regular_space_mtopology mtopology_of_def) |
|
2406 |
show "S \<subseteq> mtopology_of m1 closure_of S" |
|
2407 |
by (simp add: assms(3) closure_of_subset) |
|
2408 |
show "continuous_map (subtopology (mtopology_of m1) S) (mtopology_of m2) f" |
|
2409 |
by (metis Cauchy_continuous_imp_continuous_map f mtopology_of_submetric) |
|
2410 |
fix a |
|
2411 |
assume a: "a \<in> mtopology_of m1 closure_of S - S" |
|
2412 |
then obtain \<sigma> where ran\<sigma>: "range \<sigma> \<subseteq> S" "range \<sigma> \<subseteq> mspace m1" |
|
2413 |
and lim\<sigma>: "limitin L.M1.mtopology \<sigma> a sequentially" |
|
2414 |
by (force simp: mtopology_of_def L.M1.closure_of_sequentially) |
|
2415 |
then have "L.M1.MCauchy \<sigma>" |
|
2416 |
by (simp add: L.M1.convergent_imp_MCauchy mtopology_of_def) |
|
2417 |
then have "L.M2.MCauchy (f \<circ> \<sigma>)" |
|
2418 |
using f ran\<sigma> by (simp add: Cauchy_continuous_map_def L.M1.subspace Metric_space.MCauchy_def) |
|
2419 |
then obtain l where l: "limitin L.M2.mtopology (f \<circ> \<sigma>) l sequentially" |
|
2420 |
by (meson L.M2.mcomplete_def m2 mcomplete_of_def) |
|
2421 |
have "limitin L.M2.mtopology f l (atin_within L.M1.mtopology a S)" |
|
2422 |
unfolding L.limit_atin_sequentially_within imp_conjL |
|
2423 |
proof (intro conjI strip) |
|
2424 |
show "l \<in> mspace m2" |
|
2425 |
using L.M2.limitin_mspace l by blast |
|
2426 |
fix \<rho> |
|
2427 |
assume "range \<rho> \<subseteq> S \<inter> mspace m1 - {a}" and lim\<rho>: "limitin L.M1.mtopology \<rho> a sequentially" |
|
2428 |
then have ran\<rho>: "range \<rho> \<subseteq> S" "range \<rho> \<subseteq> mspace m1" "\<And>n. \<rho> n \<noteq> a" |
|
2429 |
by auto |
|
2430 |
have "a \<in> mspace m1" |
|
2431 |
using L.M1.limitin_mspace lim\<rho> by auto |
|
2432 |
have "S.MCauchy \<sigma>" "S.MCauchy \<rho>" |
|
2433 |
using L.M1.convergent_imp_MCauchy L.M1.MCauchy_def S.MCauchy_def lim\<sigma> ran\<sigma> lim\<rho> ran\<rho> by force+ |
|
2434 |
then have "L.M2.MCauchy (f \<circ> \<rho>)" "L.M2.MCauchy (f \<circ> \<sigma>)" |
|
2435 |
using f by (auto simp: Cauchy_continuous_map_def) |
|
2436 |
then have ran_f: "range (\<lambda>x. f (\<rho> x)) \<subseteq> mspace m2" "range (\<lambda>x. f (\<sigma> x)) \<subseteq> mspace m2" |
|
2437 |
by (auto simp: L.M2.MCauchy_def) |
|
2438 |
have "(\<lambda>n. mdist m2 (f (\<rho> n)) l) \<longlonglongrightarrow> 0" |
|
2439 |
proof (rule Lim_null_comparison) |
|
2440 |
have "mdist m2 (f (\<rho> n)) l \<le> mdist m2 (f (\<sigma> n)) l + mdist m2 (f (\<sigma> n)) (f (\<rho> n))" for n |
|
2441 |
using \<open>l \<in> mspace m2\<close> ran_f L.M2.triangle'' by (smt (verit, best) range_subsetD) |
|
2442 |
then show "\<forall>\<^sub>F n in sequentially. norm (mdist m2 (f (\<rho> n)) l) \<le> mdist m2 (f (\<sigma> n)) l + mdist m2 (f (\<sigma> n)) (f (\<rho> n))" |
|
2443 |
by force |
|
2444 |
define \<psi> where "\<psi> \<equiv> \<lambda>n. if even n then \<sigma> (n div 2) else \<rho> (n div 2)" |
|
2445 |
have "(\<lambda>n. mdist m1 (\<sigma> n) (\<rho> n)) \<longlonglongrightarrow> 0" |
|
2446 |
proof (rule Lim_null_comparison) |
|
2447 |
show "\<forall>\<^sub>F n in sequentially. norm (mdist m1 (\<sigma> n) (\<rho> n)) \<le> mdist m1 (\<sigma> n) a + mdist m1 (\<rho> n) a" |
|
2448 |
using L.M1.triangle' [of _ a] ran\<sigma> ran\<rho> \<open>a \<in> mspace m1\<close> by (simp add: range_subsetD) |
|
2449 |
have "(\<lambda>n. mdist m1 (\<sigma> n) a) \<longlonglongrightarrow> 0" |
|
2450 |
using L.M1.limitin_metric_dist_null lim\<sigma> by blast |
|
2451 |
moreover have "(\<lambda>n. mdist m1 (\<rho> n) a) \<longlonglongrightarrow> 0" |
|
2452 |
using L.M1.limitin_metric_dist_null lim\<rho> by blast |
|
2453 |
ultimately show "(\<lambda>n. mdist m1 (\<sigma> n) a + mdist m1 (\<rho> n) a) \<longlonglongrightarrow> 0" |
|
2454 |
by (simp add: tendsto_add_zero) |
|
2455 |
qed |
|
2456 |
with \<open>S.MCauchy \<sigma>\<close> \<open>S.MCauchy \<rho>\<close> have "S.MCauchy \<psi>" |
|
2457 |
by (simp add: S.MCauchy_interleaving_gen \<psi>_def) |
|
2458 |
then have "L.M2.MCauchy (f \<circ> \<psi>)" |
|
2459 |
by (metis Cauchy_continuous_map_def f mdist_submetric mspace_submetric) |
|
2460 |
then have "(\<lambda>n. mdist m2 (f (\<sigma> n)) (f (\<rho> n))) \<longlonglongrightarrow> 0" |
|
2461 |
using L.M2.MCauchy_interleaving_gen [of "f \<circ> \<sigma>" "f \<circ> \<rho>"] |
|
2462 |
by (simp add: if_distrib \<psi>_def o_def cong: if_cong) |
|
2463 |
moreover have "\<forall>\<^sub>F n in sequentially. f (\<sigma> n) \<in> mspace m2 \<and> (\<lambda>x. mdist m2 (f (\<sigma> x)) l) \<longlonglongrightarrow> 0" |
|
2464 |
using l by (auto simp: L.M2.limitin_metric_dist_null \<open>l \<in> mspace m2\<close>) |
|
2465 |
ultimately show "(\<lambda>n. mdist m2 (f (\<sigma> n)) l + mdist m2 (f (\<sigma> n)) (f (\<rho> n))) \<longlonglongrightarrow> 0" |
|
2466 |
by (metis (mono_tags) tendsto_add_zero eventually_sequentially order_refl) |
|
2467 |
qed |
|
2468 |
with ran_f show "limitin L.M2.mtopology (f \<circ> \<rho>) l sequentially" |
|
2469 |
by (auto simp: L.M2.limitin_metric_dist_null eventually_sequentially \<open>l \<in> mspace m2\<close>) |
|
2470 |
qed |
|
2471 |
then show "\<exists>l. limitin (mtopology_of m2) f l (atin_within (mtopology_of m1) a S)" |
|
2472 |
by (force simp: mtopology_of_def) |
|
2473 |
qed auto |
|
2474 |
||
2475 |
||
2476 |
lemma Cauchy_continuous_map_extends_to_continuous_closure_of: |
|
2477 |
assumes "mcomplete_of m2" |
|
2478 |
and f: "Cauchy_continuous_map (submetric m1 S) m2 f" |
|
2479 |
obtains g |
|
2480 |
where "continuous_map (subtopology (mtopology_of m1) ((mtopology_of m1) closure_of S)) |
|
2481 |
(mtopology_of m2) g" "\<And>x. x \<in> S \<Longrightarrow> g x = f x" |
|
2482 |
proof - |
|
2483 |
obtain g where cmg: |
|
2484 |
"continuous_map (subtopology (mtopology_of m1) ((mtopology_of m1) closure_of (mspace m1 \<inter> S))) |
|
2485 |
(mtopology_of m2) g" |
|
2486 |
and gf: "(\<forall>x \<in> mspace m1 \<inter> S. g x = f x)" |
|
2487 |
using Cauchy_continuous_map_extends_to_continuous_closure_of_aux assms |
|
2488 |
by (metis inf_commute inf_le2 submetric_restrict) |
|
2489 |
define h where "h \<equiv> \<lambda>x. if x \<in> topspace(mtopology_of m1) then g x else f x" |
|
2490 |
show thesis |
|
2491 |
proof |
|
2492 |
show "continuous_map (subtopology (mtopology_of m1) ((mtopology_of m1) closure_of S)) |
|
2493 |
(mtopology_of m2) h" |
|
2494 |
unfolding h_def |
|
2495 |
proof (rule continuous_map_eq) |
|
2496 |
show "continuous_map (subtopology (mtopology_of m1) (mtopology_of m1 closure_of S)) (mtopology_of m2) g" |
|
2497 |
by (metis closure_of_restrict cmg topspace_mtopology_of) |
|
2498 |
qed auto |
|
2499 |
qed (auto simp: gf h_def) |
|
2500 |
qed |
|
2501 |
||
2502 |
||
2503 |
lemma Cauchy_continuous_map_extends_to_continuous_intermediate_closure_of: |
|
2504 |
assumes "mcomplete_of m2" |
|
2505 |
and f: "Cauchy_continuous_map (submetric m1 S) m2 f" |
|
2506 |
and T: "T \<subseteq> mtopology_of m1 closure_of S" |
|
2507 |
obtains g |
|
2508 |
where "continuous_map (subtopology (mtopology_of m1) T) (mtopology_of m2) g" |
|
2509 |
"(\<forall>x \<in> S. g x = f x)" |
|
2510 |
by (metis Cauchy_continuous_map_extends_to_continuous_closure_of T assms(1) continuous_map_from_subtopology_mono f) |
|
2511 |
||
2512 |
text \<open>Technical lemma helpful for porting particularly ugly HOL Light proofs\<close> |
|
2513 |
lemma all_in_closure_of: |
|
2514 |
assumes P: "\<forall>x \<in> S. P x" and clo: "closedin X {x \<in> topspace X. P x}" |
|
2515 |
shows "\<forall>x \<in> X closure_of S. P x" |
|
2516 |
proof - |
|
2517 |
have *: "topspace X \<inter> S \<subseteq> {x \<in> topspace X. P x}" |
|
2518 |
using P by auto |
|
2519 |
show ?thesis |
|
2520 |
using closure_of_minimal [OF * clo] closure_of_restrict by fastforce |
|
2521 |
qed |
|
2522 |
||
2523 |
lemma Lipschitz_continuous_map_on_intermediate_closure_aux: |
|
2524 |
assumes lcf: "Lipschitz_continuous_map (submetric m1 S) m2 f" |
|
2525 |
and "S \<subseteq> T" and Tsub: "T \<subseteq> (mtopology_of m1) closure_of S" |
|
2526 |
and cmf: "continuous_map (subtopology (mtopology_of m1) T) (mtopology_of m2) f" |
|
2527 |
and "S \<subseteq> mspace m1" |
|
2528 |
shows "Lipschitz_continuous_map (submetric m1 T) m2 f" |
|
2529 |
proof - |
|
2530 |
interpret L: Metric_space12 "mspace m1" "mdist m1" "mspace m2" "mdist m2" |
|
2531 |
by (simp add: Metric_space12_mspace_mdist) |
|
2532 |
interpret S: Metric_space "S \<inter> mspace m1" "mdist m1" |
|
2533 |
by (simp add: L.M1.subspace) |
|
2534 |
have "T \<subseteq> mspace m1" |
|
2535 |
using Tsub by (auto simp: mtopology_of_def closure_of_def) |
|
2536 |
show ?thesis |
|
2537 |
unfolding Lipschitz_continuous_map_pos |
|
2538 |
proof |
|
78248
740b23f1138a
EXPERIMENTAL replacement of f ` A <= B by f : A -> B in Analysis
paulson <lp15@cam.ac.uk>
parents:
78202
diff
changeset
|
2539 |
show "f \<in> mspace (submetric m1 T) \<rightarrow> mspace m2" |
740b23f1138a
EXPERIMENTAL replacement of f ` A <= B by f : A -> B in Analysis
paulson <lp15@cam.ac.uk>
parents:
78202
diff
changeset
|
2540 |
by (metis cmf Metric_space.metric_continuous_map Metric_space_mspace_mdist mtopology_of_def |
740b23f1138a
EXPERIMENTAL replacement of f ` A <= B by f : A -> B in Analysis
paulson <lp15@cam.ac.uk>
parents:
78202
diff
changeset
|
2541 |
mtopology_of_submetric image_subset_iff_funcset) |
78202 | 2542 |
define X where "X \<equiv> prod_topology (subtopology L.M1.mtopology T) (subtopology L.M1.mtopology T)" |
2543 |
obtain B::real where "B > 0" and B: "\<forall>(x,y) \<in> S\<times>S. mdist m2 (f x) (f y) \<le> B * mdist m1 x y" |
|
2544 |
using lcf \<open>S \<subseteq> mspace m1\<close> by (force simp: Lipschitz_continuous_map_pos) |
|
2545 |
have eq: "{z \<in> A. case z of (x,y) \<Rightarrow> p x y \<le> B * q x y} = {z \<in> A. ((\<lambda>(x,y). B * q x y - p x y)z) \<in> {0..}}" |
|
2546 |
for p q and A::"('a*'a)set" |
|
2547 |
by auto |
|
2548 |
have clo: "closedin X {z \<in> topspace X. case z of (x, y) \<Rightarrow> mdist m2 (f x) (f y) \<le> B * mdist m1 x y}" |
|
2549 |
unfolding eq |
|
2550 |
proof (rule closedin_continuous_map_preimage) |
|
2551 |
have *: "continuous_map X L.M2.mtopology (f \<circ> fst)" "continuous_map X L.M2.mtopology (f \<circ> snd)" |
|
2552 |
using cmf by (auto simp: mtopology_of_def X_def intro: continuous_map_compose continuous_map_fst continuous_map_snd) |
|
2553 |
then show "continuous_map X euclidean (\<lambda>x. case x of (x, y) \<Rightarrow> B * mdist m1 x y - mdist m2 (f x) (f y))" |
|
2554 |
unfolding case_prod_unfold |
|
2555 |
proof (intro continuous_intros; simp add: mtopology_of_def o_def) |
|
2556 |
show "continuous_map X L.M1.mtopology fst" "continuous_map X L.M1.mtopology snd" |
|
2557 |
by (simp_all add: X_def continuous_map_subtopology_fst continuous_map_subtopology_snd flip: subtopology_Times) |
|
2558 |
qed |
|
2559 |
qed auto |
|
2560 |
have "mdist m2 (f x) (f y) \<le> B * mdist m1 x y" if "x \<in> T" "y \<in> T" for x y |
|
2561 |
using all_in_closure_of [OF B clo] \<open>S \<subseteq> T\<close> Tsub |
|
2562 |
by (fastforce simp: X_def subset_iff closure_of_Times closure_of_subtopology inf.absorb2 |
|
2563 |
mtopology_of_def that) |
|
2564 |
then show "\<exists>B>0. \<forall>x\<in>mspace (submetric m1 T). |
|
2565 |
\<forall>y\<in>mspace (submetric m1 T). |
|
2566 |
mdist m2 (f x) (f y) \<le> B * mdist (submetric m1 T) x y" |
|
2567 |
using \<open>0 < B\<close> by auto |
|
2568 |
qed |
|
2569 |
qed |
|
2570 |
||
2571 |
||
2572 |
lemma Lipschitz_continuous_map_on_intermediate_closure: |
|
2573 |
assumes "Lipschitz_continuous_map (submetric m1 S) m2 f" |
|
2574 |
and "S \<subseteq> T" "T \<subseteq> (mtopology_of m1) closure_of S" |
|
2575 |
and "continuous_map (subtopology (mtopology_of m1) T) (mtopology_of m2) f" |
|
2576 |
shows "Lipschitz_continuous_map (submetric m1 T) m2 f" |
|
2577 |
by (metis Lipschitz_continuous_map_on_intermediate_closure_aux assms closure_of_subset_topspace subset_trans topspace_mtopology_of) |
|
2578 |
||
2579 |
||
2580 |
lemma Lipschitz_continuous_map_extends_to_closure_of: |
|
2581 |
assumes m2: "mcomplete_of m2" |
|
2582 |
and f: "Lipschitz_continuous_map (submetric m1 S) m2 f" |
|
2583 |
obtains g |
|
2584 |
where "Lipschitz_continuous_map (submetric m1 (mtopology_of m1 closure_of S)) m2 g" |
|
2585 |
"\<And>x. x \<in> S \<Longrightarrow> g x = f x" |
|
2586 |
proof - |
|
2587 |
obtain g |
|
2588 |
where g: "continuous_map (subtopology (mtopology_of m1) ((mtopology_of m1) closure_of S)) |
|
2589 |
(mtopology_of m2) g" "(\<forall>x \<in> S. g x = f x)" |
|
2590 |
by (metis Cauchy_continuous_map_extends_to_continuous_closure_of Lipschitz_imp_Cauchy_continuous_map f m2) |
|
2591 |
have "Lipschitz_continuous_map (submetric m1 (mtopology_of m1 closure_of S)) m2 g" |
|
2592 |
proof (rule Lipschitz_continuous_map_on_intermediate_closure) |
|
2593 |
show "Lipschitz_continuous_map (submetric m1 (mspace m1 \<inter> S)) m2 g" |
|
2594 |
by (smt (verit, best) IntD2 Lipschitz_continuous_map_eq f g(2) inf_commute mspace_submetric submetric_restrict) |
|
2595 |
show "mspace m1 \<inter> S \<subseteq> mtopology_of m1 closure_of S" |
|
2596 |
using closure_of_subset_Int by force |
|
2597 |
show "mtopology_of m1 closure_of S \<subseteq> mtopology_of m1 closure_of (mspace m1 \<inter> S)" |
|
2598 |
by (metis closure_of_restrict subset_refl topspace_mtopology_of) |
|
2599 |
show "continuous_map (subtopology (mtopology_of m1) (mtopology_of m1 closure_of S)) (mtopology_of m2) g" |
|
2600 |
by (simp add: g) |
|
2601 |
qed |
|
2602 |
with g that show thesis |
|
2603 |
by metis |
|
2604 |
qed |
|
2605 |
||
2606 |
||
2607 |
lemma Lipschitz_continuous_map_extends_to_intermediate_closure_of: |
|
2608 |
assumes "mcomplete_of m2" |
|
2609 |
and "Lipschitz_continuous_map (submetric m1 S) m2 f" |
|
2610 |
and "T \<subseteq> mtopology_of m1 closure_of S" |
|
2611 |
obtains g |
|
2612 |
where "Lipschitz_continuous_map (submetric m1 T) m2 g" "\<And>x. x \<in> S \<Longrightarrow> g x = f x" |
|
2613 |
by (metis Lipschitz_continuous_map_extends_to_closure_of Lipschitz_continuous_map_from_submetric_mono assms) |
|
2614 |
||
2615 |
text \<open>This proof uses the same trick to extend the function's domain to its closure\<close> |
|
2616 |
lemma uniformly_continuous_map_on_intermediate_closure_aux: |
|
2617 |
assumes ucf: "uniformly_continuous_map (submetric m1 S) m2 f" |
|
2618 |
and "S \<subseteq> T" and Tsub: "T \<subseteq> (mtopology_of m1) closure_of S" |
|
2619 |
and cmf: "continuous_map (subtopology (mtopology_of m1) T) (mtopology_of m2) f" |
|
2620 |
and "S \<subseteq> mspace m1" |
|
2621 |
shows "uniformly_continuous_map (submetric m1 T) m2 f" |
|
2622 |
proof - |
|
2623 |
interpret L: Metric_space12 "mspace m1" "mdist m1" "mspace m2" "mdist m2" |
|
2624 |
by (simp add: Metric_space12_mspace_mdist) |
|
2625 |
interpret S: Metric_space "S \<inter> mspace m1" "mdist m1" |
|
2626 |
by (simp add: L.M1.subspace) |
|
2627 |
have "T \<subseteq> mspace m1" |
|
2628 |
using Tsub by (auto simp: mtopology_of_def closure_of_def) |
|
2629 |
show ?thesis |
|
2630 |
unfolding uniformly_continuous_map_def |
|
2631 |
proof (intro conjI strip) |
|
78248
740b23f1138a
EXPERIMENTAL replacement of f ` A <= B by f : A -> B in Analysis
paulson <lp15@cam.ac.uk>
parents:
78202
diff
changeset
|
2632 |
show "f \<in> mspace (submetric m1 T) \<rightarrow> mspace m2" |
740b23f1138a
EXPERIMENTAL replacement of f ` A <= B by f : A -> B in Analysis
paulson <lp15@cam.ac.uk>
parents:
78202
diff
changeset
|
2633 |
by (metis cmf Metric_space.metric_continuous_map Metric_space_mspace_mdist |
740b23f1138a
EXPERIMENTAL replacement of f ` A <= B by f : A -> B in Analysis
paulson <lp15@cam.ac.uk>
parents:
78202
diff
changeset
|
2634 |
mtopology_of_def mtopology_of_submetric image_subset_iff_funcset) |
78202 | 2635 |
fix \<epsilon>::real |
2636 |
assume "\<epsilon> > 0" |
|
2637 |
then obtain \<delta> where "\<delta>>0" and \<delta>: "\<forall>(x,y) \<in> S\<times>S. mdist m1 x y < \<delta> \<longrightarrow> mdist m2 (f x) (f y) \<le> \<epsilon>/2" |
|
2638 |
using ucf \<open>S \<subseteq> mspace m1\<close> unfolding uniformly_continuous_map_def mspace_submetric |
|
2639 |
apply (simp add: Ball_def del: divide_const_simps) |
|
2640 |
by (metis IntD2 half_gt_zero inf.orderE less_eq_real_def) |
|
2641 |
define X where "X \<equiv> prod_topology (subtopology L.M1.mtopology T) (subtopology L.M1.mtopology T)" |
|
2642 |
text \<open>A clever construction involving the union of two closed sets\<close> |
|
2643 |
have eq: "{z \<in> A. case z of (x,y) \<Rightarrow> p x y < d \<longrightarrow> q x y \<le> e} |
|
2644 |
= {z \<in> A. ((\<lambda>(x,y). p x y - d)z) \<in> {0..}} \<union> {z \<in> A. ((\<lambda>(x,y). e - q x y)z) \<in> {0..}}" |
|
2645 |
for p q and d e::real and A::"('a*'a)set" |
|
2646 |
by auto |
|
2647 |
have clo: "closedin X {z \<in> topspace X. case z of (x, y) \<Rightarrow> mdist m1 x y < \<delta> \<longrightarrow> mdist m2 (f x) (f y) \<le> \<epsilon>/2}" |
|
2648 |
unfolding eq |
|
2649 |
proof (intro closedin_Un closedin_continuous_map_preimage) |
|
2650 |
have *: "continuous_map X L.M1.mtopology fst" "continuous_map X L.M1.mtopology snd" |
|
2651 |
by (metis X_def continuous_map_subtopology_fst subtopology_Times continuous_map_subtopology_snd)+ |
|
2652 |
show "continuous_map X euclidean (\<lambda>x. case x of (x, y) \<Rightarrow> mdist m1 x y - \<delta>)" |
|
2653 |
unfolding case_prod_unfold |
|
2654 |
by (intro continuous_intros; simp add: mtopology_of_def *) |
|
2655 |
have *: "continuous_map X L.M2.mtopology (f \<circ> fst)" "continuous_map X L.M2.mtopology (f \<circ> snd)" |
|
2656 |
using cmf by (auto simp: mtopology_of_def X_def intro: continuous_map_compose continuous_map_fst continuous_map_snd) |
|
2657 |
then show "continuous_map X euclidean (\<lambda>x. case x of (x, y) \<Rightarrow> \<epsilon> / 2 - mdist m2 (f x) (f y))" |
|
2658 |
unfolding case_prod_unfold |
|
2659 |
by (intro continuous_intros; simp add: mtopology_of_def o_def) |
|
2660 |
qed auto |
|
2661 |
have "mdist m2 (f x) (f y) \<le> \<epsilon>/2" if "x \<in> T" "y \<in> T" "mdist m1 x y < \<delta>" for x y |
|
2662 |
using all_in_closure_of [OF \<delta> clo] \<open>S \<subseteq> T\<close> Tsub |
|
2663 |
by (fastforce simp: X_def subset_iff closure_of_Times closure_of_subtopology inf.absorb2 |
|
2664 |
mtopology_of_def that) |
|
2665 |
then show "\<exists>\<delta>>0. \<forall>x\<in>mspace (submetric m1 T). \<forall>y\<in>mspace (submetric m1 T). mdist (submetric m1 T) y x < \<delta> \<longrightarrow> mdist m2 (f y) (f x) < \<epsilon>" |
|
2666 |
using \<open>0 < \<delta>\<close> \<open>0 < \<epsilon>\<close> by fastforce |
|
2667 |
qed |
|
2668 |
qed |
|
2669 |
||
2670 |
lemma uniformly_continuous_map_on_intermediate_closure: |
|
2671 |
assumes "uniformly_continuous_map (submetric m1 S) m2 f" |
|
2672 |
and "S \<subseteq> T" and"T \<subseteq> (mtopology_of m1) closure_of S" |
|
2673 |
and "continuous_map (subtopology (mtopology_of m1) T) (mtopology_of m2) f" |
|
2674 |
shows "uniformly_continuous_map (submetric m1 T) m2 f" |
|
2675 |
by (metis assms closure_of_subset_topspace subset_trans topspace_mtopology_of |
|
2676 |
uniformly_continuous_map_on_intermediate_closure_aux) |
|
2677 |
||
2678 |
lemma uniformly_continuous_map_extends_to_closure_of: |
|
2679 |
assumes m2: "mcomplete_of m2" |
|
2680 |
and f: "uniformly_continuous_map (submetric m1 S) m2 f" |
|
2681 |
obtains g |
|
2682 |
where "uniformly_continuous_map (submetric m1 (mtopology_of m1 closure_of S)) m2 g" |
|
2683 |
"\<And>x. x \<in> S \<Longrightarrow> g x = f x" |
|
2684 |
proof - |
|
2685 |
obtain g |
|
2686 |
where g: "continuous_map (subtopology (mtopology_of m1) ((mtopology_of m1) closure_of S)) |
|
2687 |
(mtopology_of m2) g" "(\<forall>x \<in> S. g x = f x)" |
|
2688 |
by (metis Cauchy_continuous_map_extends_to_continuous_closure_of uniformly_imp_Cauchy_continuous_map f m2) |
|
2689 |
have "uniformly_continuous_map (submetric m1 (mtopology_of m1 closure_of S)) m2 g" |
|
2690 |
proof (rule uniformly_continuous_map_on_intermediate_closure) |
|
2691 |
show "uniformly_continuous_map (submetric m1 (mspace m1 \<inter> S)) m2 g" |
|
2692 |
by (smt (verit, best) IntD2 uniformly_continuous_map_eq f g(2) inf_commute mspace_submetric submetric_restrict) |
|
2693 |
show "mspace m1 \<inter> S \<subseteq> mtopology_of m1 closure_of S" |
|
2694 |
using closure_of_subset_Int by force |
|
2695 |
show "mtopology_of m1 closure_of S \<subseteq> mtopology_of m1 closure_of (mspace m1 \<inter> S)" |
|
2696 |
by (metis closure_of_restrict subset_refl topspace_mtopology_of) |
|
2697 |
show "continuous_map (subtopology (mtopology_of m1) (mtopology_of m1 closure_of S)) (mtopology_of m2) g" |
|
2698 |
by (simp add: g) |
|
2699 |
qed |
|
2700 |
with g that show thesis |
|
2701 |
by metis |
|
2702 |
qed |
|
2703 |
||
2704 |
||
2705 |
lemma uniformly_continuous_map_extends_to_intermediate_closure_of: |
|
2706 |
assumes "mcomplete_of m2" |
|
2707 |
and "uniformly_continuous_map (submetric m1 S) m2 f" |
|
2708 |
and "T \<subseteq> mtopology_of m1 closure_of S" |
|
2709 |
obtains g |
|
2710 |
where "uniformly_continuous_map (submetric m1 T) m2 g" "\<And>x. x \<in> S \<Longrightarrow> g x = f x" |
|
2711 |
by (metis uniformly_continuous_map_extends_to_closure_of uniformly_continuous_map_from_submetric_mono assms) |
|
2712 |
||
2713 |
||
2714 |
lemma Cauchy_continuous_map_on_intermediate_closure_aux: |
|
2715 |
assumes ucf: "Cauchy_continuous_map (submetric m1 S) m2 f" |
|
2716 |
and "S \<subseteq> T" and Tsub: "T \<subseteq> (mtopology_of m1) closure_of S" |
|
2717 |
and cmf: "continuous_map (subtopology (mtopology_of m1) T) (mtopology_of m2) f" |
|
2718 |
and "S \<subseteq> mspace m1" |
|
2719 |
shows "Cauchy_continuous_map (submetric m1 T) m2 f" |
|
2720 |
proof - |
|
2721 |
interpret L: Metric_space12 "mspace m1" "mdist m1" "mspace m2" "mdist m2" |
|
2722 |
by (simp add: Metric_space12_mspace_mdist) |
|
2723 |
interpret S: Metric_space "S \<inter> mspace m1" "mdist m1" |
|
2724 |
by (simp add: L.M1.subspace) |
|
2725 |
interpret T: Metric_space T "mdist m1" |
|
2726 |
by (metis L.M1.subspace Tsub closure_of_subset_topspace dual_order.trans topspace_mtopology_of) |
|
2727 |
have "T \<subseteq> mspace m1" |
|
2728 |
using Tsub by (auto simp: mtopology_of_def closure_of_def) |
|
2729 |
then show ?thesis |
|
2730 |
proof (clarsimp simp: Cauchy_continuous_map_def Int_absorb2) |
|
2731 |
fix \<sigma> |
|
2732 |
assume \<sigma>: "T.MCauchy \<sigma>" |
|
2733 |
have "\<exists>y\<in>S. mdist m1 (\<sigma> n) y < inverse (Suc n) \<and> mdist m2 (f (\<sigma> n)) (f y) < inverse (Suc n)" for n |
|
2734 |
proof - |
|
2735 |
have "\<sigma> n \<in> T" |
|
2736 |
using \<sigma> by (force simp: T.MCauchy_def) |
|
2737 |
moreover have "continuous_map (mtopology_of (submetric m1 T)) L.M2.mtopology f" |
|
2738 |
by (metis cmf mtopology_of_def mtopology_of_submetric) |
|
2739 |
ultimately obtain \<delta> where "\<delta>>0" and \<delta>: "\<forall>x \<in> T. mdist m1 (\<sigma> n) x < \<delta> \<longrightarrow> mdist m2 (f(\<sigma> n)) (f x) < inverse (Suc n)" |
|
2740 |
using \<open>T \<subseteq> mspace m1\<close> |
|
2741 |
apply (simp add: mtopology_of_def Metric_space.metric_continuous_map L.M1.subspace Int_absorb2) |
|
2742 |
by (metis inverse_Suc of_nat_Suc) |
|
2743 |
have "\<exists>y \<in> S. mdist m1 (\<sigma> n) y < min \<delta> (inverse (Suc n))" |
|
2744 |
using \<open>\<sigma> n \<in> T\<close> Tsub \<open>\<delta>>0\<close> |
|
2745 |
unfolding mtopology_of_def L.M1.metric_closure_of subset_iff mem_Collect_eq L.M1.in_mball |
|
2746 |
by (smt (verit, del_insts) inverse_Suc ) |
|
2747 |
with \<delta> \<open>S \<subseteq> T\<close> show ?thesis |
|
2748 |
by auto |
|
2749 |
qed |
|
2750 |
then obtain \<rho> where \<rho>S: "\<And>n. \<rho> n \<in> S" and \<rho>1: "\<And>n. mdist m1 (\<sigma> n) (\<rho> n) < inverse (Suc n)" |
|
2751 |
and \<rho>2: "\<And>n. mdist m2 (f (\<sigma> n)) (f (\<rho> n)) < inverse (Suc n)" |
|
2752 |
by metis |
|
2753 |
have "S.MCauchy \<rho>" |
|
2754 |
unfolding S.MCauchy_def |
|
2755 |
proof (intro conjI strip) |
|
2756 |
show "range \<rho> \<subseteq> S \<inter> mspace m1" |
|
2757 |
using \<open>S \<subseteq> mspace m1\<close> by (auto simp: \<rho>S) |
|
2758 |
fix \<epsilon> :: real |
|
2759 |
assume "\<epsilon>>0" |
|
2760 |
then obtain M where M: "\<And>n n'. M \<le> n \<Longrightarrow> M \<le> n' \<Longrightarrow> mdist m1 (\<sigma> n) (\<sigma> n') < \<epsilon>/2" |
|
2761 |
using \<sigma> unfolding T.MCauchy_def by (meson half_gt_zero) |
|
2762 |
have "\<forall>\<^sub>F n in sequentially. inverse (Suc n) < \<epsilon>/4" |
|
2763 |
using Archimedean_eventually_inverse \<open>0 < \<epsilon>\<close> divide_pos_pos zero_less_numeral by blast |
|
2764 |
then obtain N where N: "\<And>n. N \<le> n \<Longrightarrow> inverse (Suc n) < \<epsilon>/4" |
|
2765 |
by (meson eventually_sequentially) |
|
2766 |
have "mdist m1 (\<rho> n) (\<rho> n') < \<epsilon>" if "n \<ge> max M N" "n' \<ge> max M N" for n n' |
|
2767 |
proof - |
|
2768 |
have "mdist m1 (\<rho> n) (\<rho> n') \<le> mdist m1 (\<rho> n) (\<sigma> n) + mdist m1 (\<sigma> n) (\<rho> n')" |
|
2769 |
by (meson T.MCauchy_def T.triangle \<rho>S \<sigma> \<open>S \<subseteq> T\<close> rangeI subset_iff) |
|
2770 |
also have "\<dots> \<le> mdist m1 (\<rho> n) (\<sigma> n) + mdist m1 (\<sigma> n) (\<sigma> n') + mdist m1 (\<sigma> n') (\<rho> n')" |
|
2771 |
by (smt (verit, best) T.MCauchy_def T.triangle \<rho>S \<sigma> \<open>S \<subseteq> T\<close> in_mono rangeI) |
|
2772 |
also have "\<dots> < \<epsilon>/4 + \<epsilon>/2 + \<epsilon>/4" |
|
2773 |
using \<rho>1[of n] \<rho>1[of n'] N[of n] N[of n'] that M[of n n'] by (simp add: T.commute) |
|
2774 |
also have "... \<le> \<epsilon>" |
|
2775 |
by simp |
|
2776 |
finally show ?thesis . |
|
2777 |
qed |
|
2778 |
then show "\<exists>N. \<forall>n n'. N \<le> n \<longrightarrow> N \<le> n' \<longrightarrow> mdist m1 (\<rho> n) (\<rho> n') < \<epsilon>" |
|
2779 |
by blast |
|
2780 |
qed |
|
2781 |
then have f\<rho>: "L.M2.MCauchy (f \<circ> \<rho>)" |
|
2782 |
using ucf by (simp add: Cauchy_continuous_map_def) |
|
2783 |
show "L.M2.MCauchy (f \<circ> \<sigma>)" |
|
2784 |
unfolding L.M2.MCauchy_def |
|
2785 |
proof (intro conjI strip) |
|
2786 |
show "range (f \<circ> \<sigma>) \<subseteq> mspace m2" |
|
2787 |
using \<open>T \<subseteq> mspace m1\<close> \<sigma> cmf |
|
2788 |
apply (auto simp: ) |
|
2789 |
by (metis Metric_space.metric_continuous_map Metric_space_mspace_mdist T.MCauchy_def image_eqI inf.absorb1 mspace_submetric mtopology_of_def mtopology_of_submetric range_subsetD subset_iff) |
|
2790 |
fix \<epsilon> :: real |
|
2791 |
assume "\<epsilon>>0" |
|
2792 |
then obtain M where M: "\<And>n n'. M \<le> n \<Longrightarrow> M \<le> n' \<Longrightarrow> mdist m2 ((f \<circ> \<rho>) n) ((f \<circ> \<rho>) n') < \<epsilon>/2" |
|
2793 |
using f\<rho> unfolding L.M2.MCauchy_def by (meson half_gt_zero) |
|
2794 |
have "\<forall>\<^sub>F n in sequentially. inverse (Suc n) < \<epsilon>/4" |
|
2795 |
using Archimedean_eventually_inverse \<open>0 < \<epsilon>\<close> divide_pos_pos zero_less_numeral by blast |
|
2796 |
then obtain N where N: "\<And>n. N \<le> n \<Longrightarrow> inverse (Suc n) < \<epsilon>/4" |
|
2797 |
by (meson eventually_sequentially) |
|
2798 |
have "mdist m2 ((f \<circ> \<sigma>) n) ((f \<circ> \<sigma>) n') < \<epsilon>" if "n \<ge> max M N" "n' \<ge> max M N" for n n' |
|
2799 |
proof - |
|
2800 |
have "mdist m2 ((f \<circ> \<sigma>) n) ((f \<circ> \<sigma>) n') \<le> mdist m2 ((f \<circ> \<sigma>) n) ((f \<circ> \<rho>) n) + mdist m2 ((f \<circ> \<rho>) n) ((f \<circ> \<sigma>) n')" |
|
2801 |
by (meson L.M2.MCauchy_def \<open>range (f \<circ> \<sigma>) \<subseteq> mspace m2\<close> f\<rho> mdist_triangle rangeI subset_eq) |
|
2802 |
also have "\<dots> \<le> mdist m2 ((f \<circ> \<sigma>) n) ((f \<circ> \<rho>) n) + mdist m2 ((f \<circ> \<rho>) n) ((f \<circ> \<rho>) n') + mdist m2 ((f \<circ> \<rho>) n') ((f \<circ> \<sigma>) n')" |
|
2803 |
by (smt (verit) L.M2.MCauchy_def L.M2.triangle \<open>range (f \<circ> \<sigma>) \<subseteq> mspace m2\<close> f\<rho> range_subsetD) |
|
2804 |
also have "\<dots> < \<epsilon>/4 + \<epsilon>/2 + \<epsilon>/4" |
|
2805 |
using \<rho>2[of n] \<rho>2[of n'] N[of n] N[of n'] that M[of n n'] by (simp add: L.M2.commute) |
|
2806 |
also have "... \<le> \<epsilon>" |
|
2807 |
by simp |
|
2808 |
finally show ?thesis . |
|
2809 |
qed |
|
2810 |
then show "\<exists>N. \<forall>n n'. N \<le> n \<longrightarrow> N \<le> n' \<longrightarrow> mdist m2 ((f \<circ> \<sigma>) n) ((f \<circ> \<sigma>) n') < \<epsilon>" |
|
2811 |
by blast |
|
2812 |
qed |
|
2813 |
qed |
|
2814 |
qed |
|
2815 |
||
2816 |
lemma Cauchy_continuous_map_on_intermediate_closure: |
|
2817 |
assumes "Cauchy_continuous_map (submetric m1 S) m2 f" |
|
2818 |
and "S \<subseteq> T" and "T \<subseteq> (mtopology_of m1) closure_of S" |
|
2819 |
and "continuous_map (subtopology (mtopology_of m1) T) (mtopology_of m2) f" |
|
2820 |
shows "Cauchy_continuous_map (submetric m1 T) m2 f" |
|
2821 |
by (metis Cauchy_continuous_map_on_intermediate_closure_aux assms closure_of_subset_topspace order.trans topspace_mtopology_of) |
|
2822 |
||
2823 |
||
2824 |
lemma Cauchy_continuous_map_extends_to_closure_of: |
|
2825 |
assumes m2: "mcomplete_of m2" |
|
2826 |
and f: "Cauchy_continuous_map (submetric m1 S) m2 f" |
|
2827 |
obtains g |
|
2828 |
where "Cauchy_continuous_map (submetric m1 (mtopology_of m1 closure_of S)) m2 g" |
|
2829 |
"\<And>x. x \<in> S \<Longrightarrow> g x = f x" |
|
2830 |
proof - |
|
2831 |
obtain g |
|
2832 |
where g: "continuous_map (subtopology (mtopology_of m1) ((mtopology_of m1) closure_of S)) |
|
2833 |
(mtopology_of m2) g" "(\<forall>x \<in> S. g x = f x)" |
|
2834 |
by (metis Cauchy_continuous_map_extends_to_continuous_closure_of f m2) |
|
2835 |
have "Cauchy_continuous_map (submetric m1 (mtopology_of m1 closure_of S)) m2 g" |
|
2836 |
proof (rule Cauchy_continuous_map_on_intermediate_closure) |
|
2837 |
show "Cauchy_continuous_map (submetric m1 (mspace m1 \<inter> S)) m2 g" |
|
2838 |
by (smt (verit, best) IntD2 Cauchy_continuous_map_eq f g(2) inf_commute mspace_submetric submetric_restrict) |
|
2839 |
show "mspace m1 \<inter> S \<subseteq> mtopology_of m1 closure_of S" |
|
2840 |
using closure_of_subset_Int by force |
|
2841 |
show "mtopology_of m1 closure_of S \<subseteq> mtopology_of m1 closure_of (mspace m1 \<inter> S)" |
|
2842 |
by (metis closure_of_restrict subset_refl topspace_mtopology_of) |
|
2843 |
show "continuous_map (subtopology (mtopology_of m1) (mtopology_of m1 closure_of S)) (mtopology_of m2) g" |
|
2844 |
by (simp add: g) |
|
2845 |
qed |
|
2846 |
with g that show thesis |
|
2847 |
by metis |
|
2848 |
qed |
|
2849 |
||
2850 |
||
2851 |
lemma Cauchy_continuous_map_extends_to_intermediate_closure_of: |
|
2852 |
assumes "mcomplete_of m2" |
|
2853 |
and "Cauchy_continuous_map (submetric m1 S) m2 f" |
|
2854 |
and "T \<subseteq> mtopology_of m1 closure_of S" |
|
2855 |
obtains g |
|
2856 |
where "Cauchy_continuous_map (submetric m1 T) m2 g" "\<And>x. x \<in> S \<Longrightarrow> g x = f x" |
|
2857 |
by (metis Cauchy_continuous_map_extends_to_closure_of Cauchy_continuous_map_from_submetric_mono assms) |
|
2858 |
||
2859 |
||
2860 |
subsection\<open>Metric space of bounded functions\<close> |
|
2861 |
||
2862 |
context Metric_space |
|
2863 |
begin |
|
2864 |
||
2865 |
definition fspace :: "'b set \<Rightarrow> ('b \<Rightarrow> 'a) set" where |
|
2866 |
"fspace \<equiv> \<lambda>S. {f. f`S \<subseteq> M \<and> f \<in> extensional S \<and> mbounded (f`S)}" |
|
2867 |
||
2868 |
definition fdist :: "['b set, 'b \<Rightarrow> 'a, 'b \<Rightarrow> 'a] \<Rightarrow> real" where |
|
2869 |
"fdist \<equiv> \<lambda>S f g. if f \<in> fspace S \<and> g \<in> fspace S \<and> S \<noteq> {} |
|
2870 |
then Sup ((\<lambda>x. d (f x) (g x)) ` S) else 0" |
|
2871 |
||
2872 |
lemma fspace_empty [simp]: "fspace {} = {\<lambda>x. undefined}" |
|
2873 |
by (auto simp: fspace_def) |
|
2874 |
||
2875 |
lemma fdist_empty [simp]: "fdist {} = (\<lambda>x y. 0)" |
|
2876 |
by (auto simp: fdist_def) |
|
2877 |
||
2878 |
lemma fspace_in_M: "\<lbrakk>f \<in> fspace S; x \<in> S\<rbrakk> \<Longrightarrow> f x \<in> M" |
|
2879 |
by (auto simp: fspace_def) |
|
2880 |
||
2881 |
lemma bdd_above_dist: |
|
2882 |
assumes f: "f \<in> fspace S" and g: "g \<in> fspace S" and "S \<noteq> {}" |
|
2883 |
shows "bdd_above ((\<lambda>u. d (f u) (g u)) ` S)" |
|
2884 |
proof - |
|
2885 |
obtain a where "a \<in> S" |
|
2886 |
using \<open>S \<noteq> {}\<close> by blast |
|
2887 |
obtain B x C y where "B>0" and B: "f`S \<subseteq> mcball x B" |
|
2888 |
and "C>0" and C: "g`S \<subseteq> mcball y C" |
|
2889 |
using f g mbounded_pos by (auto simp: fspace_def) |
|
2890 |
have "d (f u) (g u) \<le> B + d x y + C" if "u\<in>S" for u |
|
2891 |
proof - |
|
2892 |
have "f u \<in> M" |
|
2893 |
by (meson B image_eqI mbounded_mcball mbounded_subset_mspace subsetD that) |
|
2894 |
have "g u \<in> M" |
|
2895 |
by (meson C image_eqI mbounded_mcball mbounded_subset_mspace subsetD that) |
|
2896 |
have "x \<in> M" "y \<in> M" |
|
2897 |
using B C that by auto |
|
2898 |
have "d (f u) (g u) \<le> d (f u) x + d x (g u)" |
|
2899 |
by (simp add: \<open>f u \<in> M\<close> \<open>g u \<in> M\<close> \<open>x \<in> M\<close> triangle) |
|
2900 |
also have "\<dots> \<le> d (f u) x + d x y + d y (g u)" |
|
2901 |
by (simp add: \<open>f u \<in> M\<close> \<open>g u \<in> M\<close> \<open>x \<in> M\<close> \<open>y \<in> M\<close> triangle) |
|
2902 |
also have "\<dots> \<le> B + d x y + C" |
|
2903 |
using B C commute that by fastforce |
|
2904 |
finally show ?thesis . |
|
2905 |
qed |
|
2906 |
then show ?thesis |
|
2907 |
by (meson bdd_above.I2) |
|
2908 |
qed |
|
2909 |
||
2910 |
||
2911 |
lemma Metric_space_funspace: "Metric_space (fspace S) (fdist S)" |
|
2912 |
proof |
|
2913 |
show *: "0 \<le> fdist S f g" for f g |
|
2914 |
by (auto simp: fdist_def intro: cSUP_upper2 [OF bdd_above_dist]) |
|
2915 |
show "fdist S f g = fdist S g f" for f g |
|
2916 |
by (auto simp: fdist_def commute) |
|
2917 |
show "(fdist S f g = 0) = (f = g)" |
|
2918 |
if fg: "f \<in> fspace S" "g \<in> fspace S" for f g |
|
2919 |
proof |
|
2920 |
assume 0: "fdist S f g = 0" |
|
2921 |
show "f = g" |
|
2922 |
proof (cases "S={}") |
|
2923 |
case True |
|
2924 |
with 0 that show ?thesis |
|
2925 |
by (simp add: fdist_def fspace_def) |
|
2926 |
next |
|
2927 |
case False |
|
2928 |
with 0 fg have Sup0: "(SUP x\<in>S. d (f x) (g x)) = 0" |
|
2929 |
by (simp add: fdist_def) |
|
2930 |
have "d (f x) (g x) = 0" if "x\<in>S" for x |
|
2931 |
by (smt (verit) False Sup0 \<open>x\<in>S\<close> bdd_above_dist [OF fg] less_cSUP_iff nonneg) |
|
2932 |
with fg show "f=g" |
|
2933 |
by (simp add: fspace_def extensionalityI image_subset_iff) |
|
2934 |
qed |
|
2935 |
next |
|
2936 |
show "f = g \<Longrightarrow> fdist S f g = 0" |
|
2937 |
using fspace_in_M [OF \<open>g \<in> fspace S\<close>] by (auto simp: fdist_def) |
|
2938 |
qed |
|
2939 |
show "fdist S f h \<le> fdist S f g + fdist S g h" |
|
2940 |
if fgh: "f \<in> fspace S" "g \<in> fspace S" "h \<in> fspace S" for f g h |
|
2941 |
proof (clarsimp simp add: fdist_def that) |
|
2942 |
assume "S \<noteq> {}" |
|
2943 |
have dfh: "d (f x) (h x) \<le> d (f x) (g x) + d (g x) (h x)" if "x\<in>S" for x |
|
2944 |
by (meson fgh fspace_in_M that triangle) |
|
2945 |
have bdd_fgh: "bdd_above ((\<lambda>x. d (f x) (g x)) ` S)" "bdd_above ((\<lambda>x. d (g x) (h x)) ` S)" |
|
2946 |
by (simp_all add: \<open>S \<noteq> {}\<close> bdd_above_dist that) |
|
2947 |
then obtain B C where B: "\<And>x. x\<in>S \<Longrightarrow> d (f x) (g x) \<le> B" and C: "\<And>x. x\<in>S \<Longrightarrow> d (g x) (h x) \<le> C" |
|
2948 |
by (auto simp: bdd_above_def) |
|
2949 |
then have "\<And>x. x\<in>S \<Longrightarrow> d (f x) (g x) + d (g x) (h x) \<le> B+C" |
|
2950 |
by force |
|
2951 |
then have bdd: "bdd_above ((\<lambda>x. d (f x) (g x) + d (g x) (h x)) ` S)" |
|
2952 |
by (auto simp: bdd_above_def) |
|
2953 |
then have "(SUP x\<in>S. d (f x) (h x)) \<le> (SUP x\<in>S. d (f x) (g x) + d (g x) (h x))" |
|
2954 |
by (metis (mono_tags, lifting) cSUP_mono \<open>S \<noteq> {}\<close> dfh) |
|
2955 |
also have "\<dots> \<le> (SUP x\<in>S. d (f x) (g x)) + (SUP x\<in>S. d (g x) (h x))" |
|
2956 |
by (simp add: \<open>S \<noteq> {}\<close> bdd cSUP_le_iff bdd_fgh add_mono cSup_upper) |
|
2957 |
finally show "(SUP x\<in>S. d (f x) (h x)) \<le> (SUP x\<in>S. d (f x) (g x)) + (SUP x\<in>S. d (g x) (h x))" . |
|
2958 |
qed |
|
2959 |
qed |
|
2960 |
||
78127
24b70433c2e8
New HOL Light material on metric spaces and topological spaces
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2961 |
end |
78202 | 2962 |
|
2963 |
||
2964 |
definition funspace where |
|
2965 |
"funspace S m \<equiv> metric (Metric_space.fspace (mspace m) (mdist m) S, |
|
2966 |
Metric_space.fdist (mspace m) (mdist m) S)" |
|
2967 |
||
2968 |
lemma mspace_funspace [simp]: |
|
2969 |
"mspace (funspace S m) = Metric_space.fspace (mspace m) (mdist m) S" |
|
2970 |
by (simp add: Metric_space.Metric_space_funspace Metric_space.mspace_metric funspace_def) |
|
2971 |
||
2972 |
lemma mdist_funspace [simp]: |
|
2973 |
"mdist (funspace S m) = Metric_space.fdist (mspace m) (mdist m) S" |
|
2974 |
by (simp add: Metric_space.Metric_space_funspace Metric_space.mdist_metric funspace_def) |
|
2975 |
||
2976 |
lemma funspace_imp_welldefined: |
|
2977 |
"\<lbrakk>f \<in> mspace (funspace S m); x \<in> S\<rbrakk> \<Longrightarrow> f x \<in> mspace m" |
|
2978 |
by (simp add: Metric_space.fspace_def subset_iff) |
|
2979 |
||
2980 |
lemma funspace_imp_extensional: |
|
2981 |
"f \<in> mspace (funspace S m) \<Longrightarrow> f \<in> extensional S" |
|
2982 |
by (simp add: Metric_space.fspace_def) |
|
2983 |
||
2984 |
lemma funspace_imp_bounded_image: |
|
2985 |
"f \<in> mspace (funspace S m) \<Longrightarrow> Metric_space.mbounded (mspace m) (mdist m) (f ` S)" |
|
2986 |
by (simp add: Metric_space.fspace_def) |
|
2987 |
||
2988 |
lemma funspace_imp_bounded: |
|
2989 |
"f \<in> mspace (funspace S m) \<Longrightarrow> S = {} \<or> (\<exists>c B. \<forall>x \<in> S. mdist m c (f x) \<le> B)" |
|
2990 |
by (auto simp: Metric_space.fspace_def Metric_space.mbounded) |
|
2991 |
||
2992 |
||
2993 |
lemma (in Metric_space) funspace_imp_bounded2: |
|
2994 |
assumes "f \<in> fspace S" "g \<in> fspace S" |
|
2995 |
obtains B where "\<And>x. x \<in> S \<Longrightarrow> d (f x) (g x) \<le> B" |
|
2996 |
proof - |
|
2997 |
have "mbounded (f ` S \<union> g ` S)" |
|
2998 |
using mbounded_Un assms by (force simp: fspace_def) |
|
2999 |
then show thesis |
|
3000 |
by (metis UnCI imageI mbounded_alt that) |
|
3001 |
qed |
|
3002 |
||
3003 |
lemma funspace_imp_bounded2: |
|
3004 |
assumes "f \<in> mspace (funspace S m)" "g \<in> mspace (funspace S m)" |
|
3005 |
obtains B where "\<And>x. x \<in> S \<Longrightarrow> mdist m (f x) (g x) \<le> B" |
|
3006 |
by (metis Metric_space_mspace_mdist assms mspace_funspace Metric_space.funspace_imp_bounded2) |
|
3007 |
||
3008 |
lemma (in Metric_space) funspace_mdist_le: |
|
3009 |
assumes fg: "f \<in> fspace S" "g \<in> fspace S" and "S \<noteq> {}" |
|
3010 |
shows "fdist S f g \<le> a \<longleftrightarrow> (\<forall>x \<in> S. d (f x) (g x) \<le> a)" |
|
3011 |
using assms bdd_above_dist [OF fg] by (simp add: fdist_def cSUP_le_iff) |
|
3012 |
||
3013 |
lemma funspace_mdist_le: |
|
3014 |
assumes "f \<in> mspace (funspace S m)" "g \<in> mspace (funspace S m)" and "S \<noteq> {}" |
|
3015 |
shows "mdist (funspace S m) f g \<le> a \<longleftrightarrow> (\<forall>x \<in> S. mdist m (f x) (g x) \<le> a)" |
|
3016 |
using assms by (simp add: Metric_space.funspace_mdist_le) |
|
3017 |
||
3018 |
||
3019 |
lemma (in Metric_space) mcomplete_funspace: |
|
3020 |
assumes "mcomplete" |
|
3021 |
shows "mcomplete_of (funspace S Self)" |
|
3022 |
proof - |
|
3023 |
interpret F: Metric_space "fspace S" "fdist S" |
|
3024 |
by (simp add: Metric_space_funspace) |
|
3025 |
show ?thesis |
|
3026 |
proof (cases "S={}") |
|
3027 |
case True |
|
3028 |
then show ?thesis |
|
3029 |
by (simp add: mcomplete_of_def mcomplete_trivial_singleton) |
|
3030 |
next |
|
3031 |
case False |
|
3032 |
show ?thesis |
|
3033 |
proof (clarsimp simp: mcomplete_of_def Metric_space.mcomplete_def) |
|
3034 |
fix \<sigma> |
|
3035 |
assume \<sigma>: "F.MCauchy \<sigma>" |
|
3036 |
then have \<sigma>M: "\<And>n x. x \<in> S \<Longrightarrow> \<sigma> n x \<in> M" |
|
3037 |
by (auto simp: F.MCauchy_def intro: fspace_in_M) |
|
3038 |
have fdist_less: "\<exists>N. \<forall>n n'. N \<le> n \<longrightarrow> N \<le> n' \<longrightarrow> fdist S (\<sigma> n) (\<sigma> n') < \<epsilon>" if "\<epsilon>>0" for \<epsilon> |
|
3039 |
using \<sigma> that by (auto simp: F.MCauchy_def) |
|
3040 |
have \<sigma>ext: "\<And>n. \<sigma> n \<in> extensional S" |
|
3041 |
using \<sigma> unfolding F.MCauchy_def by (auto simp: fspace_def) |
|
3042 |
have \<sigma>bd: "\<And>n. mbounded (\<sigma> n ` S)" |
|
3043 |
using \<sigma> unfolding F.MCauchy_def by (simp add: fspace_def image_subset_iff) |
|
3044 |
have \<sigma>in[simp]: "\<sigma> n \<in> fspace S" for n |
|
3045 |
using F.MCauchy_def \<sigma> by blast |
|
3046 |
have bd2: "\<And>n n'. \<exists>B. \<forall>x \<in> S. d (\<sigma> n x) (\<sigma> n' x) \<le> B" |
|
3047 |
using \<sigma> unfolding F.MCauchy_def by (metis range_subsetD funspace_imp_bounded2) |
|
3048 |
have sup: "\<And>n n' x0. x0 \<in> S \<Longrightarrow> d (\<sigma> n x0) (\<sigma> n' x0) \<le> Sup ((\<lambda>x. d (\<sigma> n x) (\<sigma> n' x)) ` S)" |
|
3049 |
proof (rule cSup_upper) |
|
3050 |
show "bdd_above ((\<lambda>x. d (\<sigma> n x) (\<sigma> n' x)) ` S)" if "x0 \<in> S" for n n' x0 |
|
3051 |
using that bd2 by (meson bdd_above.I2) |
|
3052 |
qed auto |
|
3053 |
have pcy: "MCauchy (\<lambda>n. \<sigma> n x)" if "x \<in> S" for x |
|
3054 |
unfolding MCauchy_def |
|
3055 |
proof (intro conjI strip) |
|
3056 |
show "range (\<lambda>n. \<sigma> n x) \<subseteq> M" |
|
3057 |
using \<sigma>M that by blast |
|
3058 |
fix \<epsilon> :: real |
|
3059 |
assume "\<epsilon> > 0" |
|
3060 |
then obtain N where N: "\<And>n n'. N \<le> n \<longrightarrow> N \<le> n' \<longrightarrow> fdist S (\<sigma> n) (\<sigma> n') < \<epsilon>" |
|
3061 |
using \<sigma> by (force simp: F.MCauchy_def) |
|
3062 |
{ fix n n' |
|
3063 |
assume n: "N \<le> n" "N \<le> n'" |
|
3064 |
have "d (\<sigma> n x) (\<sigma> n' x) \<le> (SUP x\<in>S. d (\<sigma> n x) (\<sigma> n' x))" |
|
3065 |
using that sup by presburger |
|
3066 |
then have "d (\<sigma> n x) (\<sigma> n' x) \<le> fdist S (\<sigma> n) (\<sigma> n')" |
|
3067 |
by (simp add: fdist_def \<open>S \<noteq> {}\<close>) |
|
3068 |
with N n have "d (\<sigma> n x) (\<sigma> n' x) < \<epsilon>" |
|
3069 |
by fastforce |
|
3070 |
} then show "\<exists>N. \<forall>n n'. N \<le> n \<longrightarrow> N \<le> n' \<longrightarrow> d (\<sigma> n x) (\<sigma> n' x) < \<epsilon>" |
|
3071 |
by blast |
|
3072 |
qed |
|
3073 |
have "\<exists>l. limitin mtopology (\<lambda>n. \<sigma> n x) l sequentially" if "x \<in> S" for x |
|
3074 |
using assms mcomplete_def pcy \<open>x \<in> S\<close> by presburger |
|
3075 |
then obtain g0 where g0: "\<And>x. x \<in> S \<Longrightarrow> limitin mtopology (\<lambda>n. \<sigma> n x) (g0 x) sequentially" |
|
3076 |
by metis |
|
3077 |
define g where "g \<equiv> restrict g0 S" |
|
3078 |
have gext: "g \<in> extensional S" |
|
3079 |
and glim: "\<And>x. x \<in> S \<Longrightarrow> limitin mtopology (\<lambda>n. \<sigma> n x) (g x) sequentially" |
|
3080 |
by (auto simp: g_def g0) |
|
3081 |
have gwd: "g x \<in> M" if "x \<in> S" for x |
|
3082 |
using glim limitin_metric that by blast |
|
3083 |
have unif: "\<exists>N. \<forall>x n. x \<in> S \<longrightarrow> N \<le> n \<longrightarrow> d (\<sigma> n x) (g x) < \<epsilon>" if "\<epsilon>>0" for \<epsilon> |
|
3084 |
proof - |
|
3085 |
obtain N where N: "\<And>n n'. N \<le> n \<and> N \<le> n' \<Longrightarrow> Sup ((\<lambda>x. d (\<sigma> n x) (\<sigma> n' x)) ` S) < \<epsilon>/2" |
|
3086 |
using \<open>S\<noteq>{}\<close> \<open>\<epsilon>>0\<close> fdist_less [of "\<epsilon>/2"] |
|
3087 |
by (metis (mono_tags) \<sigma>in fdist_def half_gt_zero) |
|
3088 |
show ?thesis |
|
3089 |
proof (intro exI strip) |
|
3090 |
fix x n |
|
3091 |
assume "x \<in> S" and "N \<le> n" |
|
3092 |
obtain N' where N': "\<And>n. N' \<le> n \<Longrightarrow> \<sigma> n x \<in> M \<and> d (\<sigma> n x) (g x) < \<epsilon>/2" |
|
3093 |
by (metis \<open>0 < \<epsilon>\<close> \<open>x \<in> S\<close> glim half_gt_zero limit_metric_sequentially) |
|
3094 |
have "d (\<sigma> n x) (g x) \<le> d (\<sigma> n x) (\<sigma> (max N N') x) + d (\<sigma> (max N N') x) (g x)" |
|
3095 |
using \<open>x \<in> S\<close> \<sigma>M gwd triangle by presburger |
|
3096 |
also have "\<dots> < \<epsilon>/2 + \<epsilon>/2" |
|
3097 |
by (smt (verit) N N' \<open>N \<le> n\<close> \<open>x \<in> S\<close> max.cobounded1 max.cobounded2 sup) |
|
3098 |
finally show "d (\<sigma> n x) (g x) < \<epsilon>" by simp |
|
3099 |
qed |
|
3100 |
qed |
|
3101 |
have "limitin F.mtopology \<sigma> g sequentially" |
|
3102 |
unfolding F.limit_metric_sequentially |
|
3103 |
proof (intro conjI strip) |
|
3104 |
obtain N where N: "\<And>n n'. N \<le> n \<and> N \<le> n' \<Longrightarrow> Sup ((\<lambda>x. d (\<sigma> n x) (\<sigma> n' x)) ` S) < 1" |
|
3105 |
using fdist_less [of 1] \<open>S\<noteq>{}\<close> by (auto simp: fdist_def) |
|
3106 |
have "\<And>x. x \<in> \<sigma> N ` S \<Longrightarrow> x \<in> M" |
|
3107 |
using \<sigma>M by blast |
|
3108 |
obtain a B where "a \<in> M" and B: "\<And>x. x \<in> (\<sigma> N) ` S \<Longrightarrow> d a x \<le> B" |
|
3109 |
by (metis False \<sigma>M \<sigma>bd ex_in_conv imageI mbounded_alt_pos) |
|
3110 |
have "d a (g x) \<le> B+1" if "x\<in>S" for x |
|
3111 |
proof - |
|
3112 |
have "d a (g x) \<le> d a (\<sigma> N x) + d (\<sigma> N x) (g x)" |
|
3113 |
by (simp add: \<open>a \<in> M\<close> \<sigma>M gwd that triangle) |
|
3114 |
also have "\<dots> \<le> B+1" |
|
3115 |
proof - |
|
3116 |
have "d a (\<sigma> N x) \<le> B" |
|
3117 |
by (simp add: B that) |
|
3118 |
moreover |
|
3119 |
have False if 1: "d (\<sigma> N x) (g x) > 1" |
|
3120 |
proof - |
|
3121 |
obtain r where "1 < r" and r: "r < d (\<sigma> N x) (g x)" |
|
3122 |
using 1 dense by blast |
|
3123 |
then obtain N' where N': "\<And>n. N' \<le> n \<Longrightarrow> \<sigma> n x \<in> M \<and> d (\<sigma> n x) (g x) < r-1" |
|
3124 |
using glim [OF \<open>x\<in>S\<close>] by (fastforce simp: limit_metric_sequentially) |
|
3125 |
have "d (\<sigma> N x) (g x) \<le> d (\<sigma> N x) (\<sigma> (max N N') x) + d (\<sigma> (max N N') x) (g x)" |
|
3126 |
by (metis \<open>x \<in> S\<close> \<sigma>M commute gwd triangle') |
|
3127 |
also have "\<dots> < 1 + (r-1)" |
|
3128 |
by (smt (verit) N N' \<open>x \<in> S\<close> max.cobounded1 max.cobounded2 max.idem sup) |
|
3129 |
finally have "d (\<sigma> N x) (g x) < r" |
|
3130 |
by simp |
|
3131 |
with r show False |
|
3132 |
by linarith |
|
3133 |
qed |
|
3134 |
ultimately show ?thesis |
|
3135 |
by force |
|
3136 |
qed |
|
3137 |
finally show ?thesis . |
|
3138 |
qed |
|
3139 |
with gwd \<open>a \<in> M\<close> have "mbounded (g ` S)" |
|
3140 |
unfolding mbounded by blast |
|
3141 |
with gwd gext show "g \<in> fspace S" |
|
3142 |
by (auto simp: fspace_def) |
|
3143 |
fix \<epsilon>::real |
|
3144 |
assume "\<epsilon>>0" |
|
3145 |
then obtain N where "\<And>x n. x \<in> S \<Longrightarrow> N \<le> n \<Longrightarrow> d (\<sigma> n x) (g x) < \<epsilon>/2" |
|
3146 |
by (meson unif half_gt_zero) |
|
3147 |
then have "fdist S (\<sigma> n) g \<le> \<epsilon>/2" if "N \<le> n" for n |
|
3148 |
using \<open>g \<in> fspace S\<close> False that |
|
3149 |
by (force simp: funspace_mdist_le simp del: divide_const_simps) |
|
3150 |
then show "\<exists>N. \<forall>n\<ge>N. \<sigma> n \<in> fspace S \<and> fdist S (\<sigma> n) g < \<epsilon>" |
|
3151 |
by (metis \<open>0 < \<epsilon>\<close> \<sigma>in add_strict_increasing field_sum_of_halves half_gt_zero) |
|
3152 |
qed |
|
3153 |
then show "\<exists>x. limitin F.mtopology \<sigma> x sequentially" |
|
3154 |
by blast |
|
3155 |
qed |
|
3156 |
qed |
|
3157 |
qed |
|
3158 |
||
3159 |
||
3160 |
||
3161 |
subsection\<open>Metric space of continuous bounded functions\<close> |
|
3162 |
||
3163 |
definition cfunspace where |
|
3164 |
"cfunspace X m \<equiv> submetric (funspace (topspace X) m) {f. continuous_map X (mtopology_of m) f}" |
|
3165 |
||
3166 |
lemma mspace_cfunspace [simp]: |
|
3167 |
"mspace (cfunspace X m) = |
|
78248
740b23f1138a
EXPERIMENTAL replacement of f ` A <= B by f : A -> B in Analysis
paulson <lp15@cam.ac.uk>
parents:
78202
diff
changeset
|
3168 |
{f. f \<in> topspace X \<rightarrow> mspace m \<and> f \<in> extensional (topspace X) \<and> |
78202 | 3169 |
Metric_space.mbounded (mspace m) (mdist m) (f ` (topspace X)) \<and> |
3170 |
continuous_map X (mtopology_of m) f}" |
|
3171 |
by (auto simp: cfunspace_def Metric_space.fspace_def) |
|
3172 |
||
3173 |
lemma mdist_cfunspace_eq_mdist_funspace: |
|
3174 |
"mdist (cfunspace X m) = mdist (funspace (topspace X) m)" |
|
3175 |
by (auto simp: cfunspace_def) |
|
3176 |
||
3177 |
lemma cfunspace_subset_funspace: |
|
3178 |
"mspace (cfunspace X m) \<subseteq> mspace (funspace (topspace X) m)" |
|
3179 |
by (simp add: cfunspace_def) |
|
3180 |
||
3181 |
lemma cfunspace_mdist_le: |
|
3182 |
"\<lbrakk>f \<in> mspace (cfunspace X m); g \<in> mspace (cfunspace X m); topspace X \<noteq> {}\<rbrakk> |
|
3183 |
\<Longrightarrow> mdist (cfunspace X m) f g \<le> a \<longleftrightarrow> (\<forall>x \<in> topspace X. mdist m (f x) (g x) \<le> a)" |
|
3184 |
by (simp add: cfunspace_def Metric_space.funspace_mdist_le) |
|
3185 |
||
3186 |
lemma cfunspace_imp_bounded2: |
|
3187 |
assumes "f \<in> mspace (cfunspace X m)" "g \<in> mspace (cfunspace X m)" |
|
3188 |
obtains B where "\<And>x. x \<in> topspace X \<Longrightarrow> mdist m (f x) (g x) \<le> B" |
|
3189 |
by (metis assms all_not_in_conv cfunspace_mdist_le nle_le) |
|
3190 |
||
3191 |
lemma cfunspace_mdist_lt: |
|
3192 |
"\<lbrakk>compactin X (topspace X); f \<in> mspace (cfunspace X m); |
|
3193 |
g \<in> mspace (cfunspace X m); mdist (cfunspace X m) f g < a; |
|
3194 |
x \<in> topspace X\<rbrakk> |
|
3195 |
\<Longrightarrow> mdist m (f x) (g x) < a" |
|
3196 |
by (metis (full_types) cfunspace_mdist_le empty_iff less_eq_real_def less_le_not_le) |
|
3197 |
||
3198 |
lemma mdist_cfunspace_le: |
|
3199 |
assumes "0 \<le> B" and B: "\<And>x. x \<in> topspace X \<Longrightarrow> mdist m (f x) (g x) \<le> B" |
|
3200 |
shows "mdist (cfunspace X m) f g \<le> B" |
|
78336 | 3201 |
proof (cases "X = trivial_topology") |
78202 | 3202 |
case True |
3203 |
then show ?thesis |
|
3204 |
by (simp add: Metric_space.fdist_empty \<open>B\<ge>0\<close> cfunspace_def) |
|
3205 |
next |
|
3206 |
case False |
|
3207 |
have bdd: "bdd_above ((\<lambda>u. mdist m (f u) (g u)) ` topspace X)" |
|
3208 |
by (meson B bdd_above.I2) |
|
3209 |
with assms bdd show ?thesis |
|
3210 |
by (simp add: mdist_cfunspace_eq_mdist_funspace Metric_space.fdist_def cSUP_le_iff) |
|
3211 |
qed |
|
3212 |
||
3213 |
||
3214 |
lemma mdist_cfunspace_imp_mdist_le: |
|
3215 |
"\<lbrakk>f \<in> mspace (cfunspace X m); g \<in> mspace (cfunspace X m); |
|
3216 |
mdist (cfunspace X m) f g \<le> a; x \<in> topspace X\<rbrakk> \<Longrightarrow> mdist m (f x) (g x) \<le> a" |
|
3217 |
using cfunspace_mdist_le by blast |
|
3218 |
||
3219 |
lemma compactin_mspace_cfunspace: |
|
3220 |
"compactin X (topspace X) |
|
3221 |
\<Longrightarrow> mspace (cfunspace X m) = |
|
3222 |
{f. (\<forall>x \<in> topspace X. f x \<in> mspace m) \<and> |
|
3223 |
f \<in> extensional (topspace X) \<and> |
|
3224 |
continuous_map X (mtopology_of m) f}" |
|
3225 |
by (auto simp: Metric_space.compactin_imp_mbounded image_compactin mtopology_of_def) |
|
3226 |
||
3227 |
lemma (in Metric_space) mcomplete_cfunspace: |
|
3228 |
assumes "mcomplete" |
|
3229 |
shows "mcomplete_of (cfunspace X Self)" |
|
3230 |
proof - |
|
3231 |
interpret F: Metric_space "fspace (topspace X)" "fdist (topspace X)" |
|
3232 |
by (simp add: Metric_space_funspace) |
|
3233 |
interpret S: Submetric "fspace (topspace X)" "fdist (topspace X)" "mspace (cfunspace X Self)" |
|
3234 |
proof |
|
3235 |
show "mspace (cfunspace X Self) \<subseteq> fspace (topspace X)" |
|
3236 |
by (metis cfunspace_subset_funspace mdist_Self mspace_Self mspace_funspace) |
|
3237 |
qed |
|
3238 |
show ?thesis |
|
78336 | 3239 |
proof (cases "X = trivial_topology") |
78202 | 3240 |
case True |
3241 |
then show ?thesis |
|
3242 |
by (simp add: mcomplete_of_def mcomplete_trivial_singleton mdist_cfunspace_eq_mdist_funspace cong: conj_cong) |
|
3243 |
next |
|
3244 |
case False |
|
3245 |
have *: "continuous_map X mtopology g" |
|
3246 |
if "range \<sigma> \<subseteq> mspace (cfunspace X Self)" |
|
3247 |
and g: "limitin F.mtopology \<sigma> g sequentially" for \<sigma> g |
|
3248 |
unfolding continuous_map_to_metric |
|
3249 |
proof (intro strip) |
|
3250 |
have \<sigma>: "\<And>n. continuous_map X mtopology (\<sigma> n)" |
|
3251 |
using that by (auto simp: mtopology_of_def) |
|
3252 |
fix x and \<epsilon>::real |
|
3253 |
assume "x \<in> topspace X" and "0 < \<epsilon>" |
|
3254 |
then obtain N where N: "\<And>n. N \<le> n \<Longrightarrow> \<sigma> n \<in> fspace (topspace X) \<and> fdist (topspace X) (\<sigma> n) g < \<epsilon>/3" |
|
3255 |
unfolding mtopology_of_def F.limitin_metric |
|
3256 |
by (metis F.limit_metric_sequentially divide_pos_pos g zero_less_numeral) |
|
3257 |
then obtain U where "openin X U" "x \<in> U" |
|
3258 |
and U: "\<And>y. y \<in> U \<Longrightarrow> \<sigma> N y \<in> mball (\<sigma> N x) (\<epsilon>/3)" |
|
3259 |
by (metis Metric_space.continuous_map_to_metric Metric_space_axioms \<open>0 < \<epsilon>\<close> \<open>x \<in> topspace X\<close> \<sigma> divide_pos_pos zero_less_numeral) |
|
3260 |
moreover |
|
3261 |
have "g y \<in> mball (g x) \<epsilon>" if "y\<in>U" for y |
|
3262 |
proof - |
|
3263 |
have "U \<subseteq> topspace X" |
|
3264 |
using \<open>openin X U\<close> by (simp add: openin_subset) |
|
3265 |
have gx: "g x \<in> M" |
|
3266 |
by (meson F.limitin_mspace \<open>x \<in> topspace X\<close> fspace_in_M g) |
|
3267 |
have "y \<in> topspace X" |
|
3268 |
using \<open>U \<subseteq> topspace X\<close> that by auto |
|
3269 |
have gy: "g y \<in> M" |
|
3270 |
by (meson F.limitin_mspace[OF g] \<open>U \<subseteq> topspace X\<close> fspace_in_M subsetD that) |
|
3271 |
have "d (g x) (g y) < \<epsilon>" |
|
3272 |
proof - |
|
3273 |
have *: "d (\<sigma> N x0) (g x0) \<le> \<epsilon>/3" if "x0 \<in> topspace X" for x0 |
|
3274 |
proof - |
|
78336 | 3275 |
have "g \<in> fspace (topspace X)" |
3276 |
using F.limit_metric_sequentially g by blast |
|
3277 |
with N that have "bdd_above ((\<lambda>x. d (\<sigma> N x) (g x)) ` topspace X)" |
|
3278 |
by (force intro: bdd_above_dist) |
|
78202 | 3279 |
then have "d (\<sigma> N x0) (g x0) \<le> Sup ((\<lambda>x. d (\<sigma> N x) (g x)) ` topspace X)" |
3280 |
by (simp add: cSup_upper that) |
|
3281 |
also have "\<dots> \<le> \<epsilon>/3" |
|
78336 | 3282 |
using g False N \<open>g \<in> fspace (topspace X)\<close> |
3283 |
by (fastforce simp: F.limit_metric_sequentially fdist_def) |
|
78202 | 3284 |
finally show ?thesis . |
3285 |
qed |
|
3286 |
have "d (g x) (g y) \<le> d (g x) (\<sigma> N x) + d (\<sigma> N x) (g y)" |
|
3287 |
using U gx gy that triangle by force |
|
3288 |
also have "\<dots> < \<epsilon>/3 + \<epsilon>/3 + \<epsilon>/3" |
|
3289 |
by (smt (verit) "*" U gy \<open>x \<in> topspace X\<close> \<open>y \<in> topspace X\<close> commute in_mball that triangle) |
|
3290 |
finally show ?thesis by simp |
|
3291 |
qed |
|
3292 |
with gx gy show ?thesis by simp |
|
3293 |
qed |
|
3294 |
ultimately show "\<exists>U. openin X U \<and> x \<in> U \<and> (\<forall>y\<in>U. g y \<in> mball (g x) \<epsilon>)" |
|
3295 |
by blast |
|
3296 |
qed |
|
3297 |
||
3298 |
have "S.sub.mcomplete" |
|
3299 |
proof (rule S.sequentially_closedin_mcomplete_imp_mcomplete) |
|
3300 |
show "F.mcomplete" |
|
3301 |
by (metis assms mcomplete_funspace mcomplete_of_def mdist_Self mdist_funspace mspace_Self mspace_funspace) |
|
3302 |
fix \<sigma> g |
|
3303 |
assume g: "range \<sigma> \<subseteq> mspace (cfunspace X Self) \<and> limitin F.mtopology \<sigma> g sequentially" |
|
3304 |
show "g \<in> mspace (cfunspace X Self)" |
|
3305 |
proof (simp add: mtopology_of_def, intro conjI) |
|
78248
740b23f1138a
EXPERIMENTAL replacement of f ` A <= B by f : A -> B in Analysis
paulson <lp15@cam.ac.uk>
parents:
78202
diff
changeset
|
3306 |
show "g \<in> topspace X \<rightarrow> M" "g \<in> extensional (topspace X)" "mbounded (g ` topspace X)" |
78202 | 3307 |
using g F.limitin_mspace by (force simp: fspace_def)+ |
3308 |
show "continuous_map X mtopology g" |
|
3309 |
using * g by blast |
|
3310 |
qed |
|
3311 |
qed |
|
3312 |
then show ?thesis |
|
3313 |
by (simp add: mcomplete_of_def mdist_cfunspace_eq_mdist_funspace) |
|
3314 |
qed |
|
3315 |
qed |
|
3316 |
||
3317 |
||
3318 |
subsection\<open>Existence of completion for any metric space M as a subspace of M=>R\<close> |
|
3319 |
||
3320 |
lemma (in Metric_space) metric_completion_explicit: |
|
3321 |
obtains f :: "['a,'a] \<Rightarrow> real" and S where |
|
3322 |
"S \<subseteq> mspace(funspace M euclidean_metric)" |
|
3323 |
"mcomplete_of (submetric (funspace M euclidean_metric) S)" |
|
78248
740b23f1138a
EXPERIMENTAL replacement of f ` A <= B by f : A -> B in Analysis
paulson <lp15@cam.ac.uk>
parents:
78202
diff
changeset
|
3324 |
"f \<in> M \<rightarrow> S" |
78202 | 3325 |
"mtopology_of(funspace M euclidean_metric) closure_of f ` M = S" |
3326 |
"\<And>x y. \<lbrakk>x \<in> M; y \<in> M\<rbrakk> |
|
3327 |
\<Longrightarrow> mdist (funspace M euclidean_metric) (f x) (f y) = d x y" |
|
3328 |
proof - |
|
3329 |
define m':: "('a\<Rightarrow>real) metric" where "m' \<equiv> funspace M euclidean_metric" |
|
3330 |
show thesis |
|
3331 |
proof (cases "M = {}") |
|
3332 |
case True |
|
3333 |
then show ?thesis |
|
3334 |
using that by (simp add: mcomplete_of_def mcomplete_trivial) |
|
3335 |
next |
|
3336 |
case False |
|
3337 |
then obtain a where "a \<in> M" |
|
3338 |
by auto |
|
3339 |
define f where "f \<equiv> \<lambda>x. (\<lambda>u \<in> M. d x u - d a u)" |
|
3340 |
define S where "S \<equiv> mtopology_of(funspace M euclidean_metric) closure_of (f ` M)" |
|
3341 |
interpret S: Submetric "Met_TC.fspace M" "Met_TC.fdist M" "S \<inter> Met_TC.fspace M" |
|
3342 |
by (simp add: Met_TC.Metric_space_funspace Submetric.intro Submetric_axioms_def) |
|
3343 |
||
3344 |
have fim: "f ` M \<subseteq> mspace m'" |
|
3345 |
proof (clarsimp simp: m'_def Met_TC.fspace_def) |
|
3346 |
fix b |
|
3347 |
assume "b \<in> M" |
|
3348 |
then have "\<And>c. \<lbrakk>c \<in> M\<rbrakk> \<Longrightarrow> \<bar>d b c - d a c\<bar> \<le> d a b" |
|
3349 |
by (smt (verit, best) \<open>a \<in> M\<close> commute triangle'') |
|
3350 |
then have "(\<lambda>x. d b x - d a x) ` M \<subseteq> cball 0 (d a b)" |
|
3351 |
by force |
|
3352 |
then show "f b \<in> extensional M \<and> bounded (f b ` M)" |
|
3353 |
by (metis bounded_cball bounded_subset f_def image_restrict_eq restrict_extensional_sub set_eq_subset) |
|
3354 |
qed |
|
3355 |
show thesis |
|
3356 |
proof |
|
3357 |
show "S \<subseteq> mspace (funspace M euclidean_metric)" |
|
3358 |
by (simp add: S_def in_closure_of subset_iff) |
|
3359 |
have "closedin S.mtopology (S \<inter> Met_TC.fspace M)" |
|
3360 |
by (simp add: S_def closedin_Int funspace_def) |
|
3361 |
moreover have "S.mcomplete" |
|
3362 |
using Metric_space.mcomplete_funspace Met_TC.Metric_space_axioms by (fastforce simp: mcomplete_of_def) |
|
3363 |
ultimately show "mcomplete_of (submetric (funspace M euclidean_metric) S)" |
|
3364 |
by (simp add: S.closedin_eq_mcomplete mcomplete_of_def) |
|
78248
740b23f1138a
EXPERIMENTAL replacement of f ` A <= B by f : A -> B in Analysis
paulson <lp15@cam.ac.uk>
parents:
78202
diff
changeset
|
3365 |
show "f \<in> M \<rightarrow> S" |
78202 | 3366 |
using S_def fim in_closure_of m'_def by fastforce |
3367 |
show "mtopology_of (funspace M euclidean_metric) closure_of f ` M = S" |
|
3368 |
by (auto simp: f_def S_def mtopology_of_def) |
|
3369 |
show "mdist (funspace M euclidean_metric) (f x) (f y) = d x y" |
|
3370 |
if "x \<in> M" "y \<in> M" for x y |
|
3371 |
proof - |
|
3372 |
have "\<forall>c\<in>M. dist (f x c) (f y c) \<le> r \<Longrightarrow> d x y \<le> r" for r |
|
3373 |
using that by (auto simp: f_def dist_real_def) |
|
3374 |
moreover have "dist (f x z) (f y z) \<le> r" if "d x y \<le> r" and "z \<in> M" for r z |
|
3375 |
using that \<open>x \<in> M\<close> \<open>y \<in> M\<close> |
|
3376 |
apply (simp add: f_def Met_TC.fdist_def dist_real_def) |
|
3377 |
by (smt (verit, best) commute triangle') |
|
3378 |
ultimately have "(SUP c \<in> M. dist (f x c) (f y c)) = d x y" |
|
3379 |
by (intro cSup_unique) auto |
|
3380 |
with that fim show ?thesis |
|
3381 |
using that fim by (simp add: Met_TC.fdist_def False m'_def image_subset_iff) |
|
3382 |
qed |
|
3383 |
qed |
|
3384 |
qed |
|
3385 |
qed |
|
3386 |
||
3387 |
||
3388 |
lemma (in Metric_space) metric_completion: |
|
3389 |
obtains f :: "['a,'a] \<Rightarrow> real" and m' where |
|
3390 |
"mcomplete_of m'" |
|
78248
740b23f1138a
EXPERIMENTAL replacement of f ` A <= B by f : A -> B in Analysis
paulson <lp15@cam.ac.uk>
parents:
78202
diff
changeset
|
3391 |
"f \<in> M \<rightarrow> mspace m' " |
78202 | 3392 |
"mtopology_of m' closure_of f ` M = mspace m'" |
3393 |
"\<And>x y. \<lbrakk>x \<in> M; y \<in> M\<rbrakk> \<Longrightarrow> mdist m' (f x) (f y) = d x y" |
|
3394 |
proof - |
|
3395 |
obtain f :: "['a,'a] \<Rightarrow> real" and S where |
|
3396 |
Ssub: "S \<subseteq> mspace(funspace M euclidean_metric)" |
|
3397 |
and mcom: "mcomplete_of (submetric (funspace M euclidean_metric) S)" |
|
78248
740b23f1138a
EXPERIMENTAL replacement of f ` A <= B by f : A -> B in Analysis
paulson <lp15@cam.ac.uk>
parents:
78202
diff
changeset
|
3398 |
and fim: "f \<in> M \<rightarrow> S" |
78202 | 3399 |
and eqS: "mtopology_of(funspace M euclidean_metric) closure_of f ` M = S" |
3400 |
and eqd: "\<And>x y. \<lbrakk>x \<in> M; y \<in> M\<rbrakk> \<Longrightarrow> mdist (funspace M euclidean_metric) (f x) (f y) = d x y" |
|
3401 |
using metric_completion_explicit by metis |
|
3402 |
define m' where "m' \<equiv> submetric (funspace M euclidean_metric) S" |
|
3403 |
show thesis |
|
3404 |
proof |
|
3405 |
show "mcomplete_of m'" |
|
3406 |
by (simp add: mcom m'_def) |
|
78248
740b23f1138a
EXPERIMENTAL replacement of f ` A <= B by f : A -> B in Analysis
paulson <lp15@cam.ac.uk>
parents:
78202
diff
changeset
|
3407 |
show "f \<in> M \<rightarrow> mspace m'" |
78202 | 3408 |
using Ssub fim m'_def by auto |
3409 |
show "mtopology_of m' closure_of f ` M = mspace m'" |
|
3410 |
using eqS fim Ssub |
|
78248
740b23f1138a
EXPERIMENTAL replacement of f ` A <= B by f : A -> B in Analysis
paulson <lp15@cam.ac.uk>
parents:
78202
diff
changeset
|
3411 |
by (force simp: m'_def mtopology_of_submetric closure_of_subtopology Int_absorb1 image_subset_iff_funcset) |
78202 | 3412 |
show "mdist m' (f x) (f y) = d x y" if "x \<in> M" and "y \<in> M" for x y |
3413 |
using that eqd m'_def by force |
|
3414 |
qed |
|
3415 |
qed |
|
3416 |
||
3417 |
lemma metrizable_space_completion: |
|
3418 |
assumes "metrizable_space X" |
|
3419 |
obtains f :: "['a,'a] \<Rightarrow> real" and Y where |
|
3420 |
"completely_metrizable_space Y" "embedding_map X Y f" |
|
3421 |
"Y closure_of (f ` (topspace X)) = topspace Y" |
|
3422 |
proof - |
|
3423 |
obtain M d where "Metric_space M d" and Xeq: "X = Metric_space.mtopology M d" |
|
3424 |
using assms metrizable_space_def by blast |
|
3425 |
then interpret Metric_space M d by simp |
|
3426 |
obtain f :: "['a,'a] \<Rightarrow> real" and m' where |
|
3427 |
"mcomplete_of m'" |
|
78248
740b23f1138a
EXPERIMENTAL replacement of f ` A <= B by f : A -> B in Analysis
paulson <lp15@cam.ac.uk>
parents:
78202
diff
changeset
|
3428 |
and fim: "f \<in> M \<rightarrow> mspace m' " |
78202 | 3429 |
and m': "mtopology_of m' closure_of f ` M = mspace m'" |
3430 |
and eqd: "\<And>x y. \<lbrakk>x \<in> M; y \<in> M\<rbrakk> \<Longrightarrow> mdist m' (f x) (f y) = d x y" |
|
78248
740b23f1138a
EXPERIMENTAL replacement of f ` A <= B by f : A -> B in Analysis
paulson <lp15@cam.ac.uk>
parents:
78202
diff
changeset
|
3431 |
by (metis metric_completion) |
78202 | 3432 |
show thesis |
3433 |
proof |
|
3434 |
show "completely_metrizable_space (mtopology_of m')" |
|
3435 |
using \<open>mcomplete_of m'\<close> |
|
3436 |
unfolding completely_metrizable_space_def mcomplete_of_def mtopology_of_def |
|
3437 |
by (metis Metric_space_mspace_mdist) |
|
3438 |
show "embedding_map X (mtopology_of m') f" |
|
3439 |
using Metric_space12.isometry_imp_embedding_map |
|
78248
740b23f1138a
EXPERIMENTAL replacement of f ` A <= B by f : A -> B in Analysis
paulson <lp15@cam.ac.uk>
parents:
78202
diff
changeset
|
3440 |
by (metis Metric_space12_def Metric_space_axioms Metric_space_mspace_mdist Xeq eqd fim |
740b23f1138a
EXPERIMENTAL replacement of f ` A <= B by f : A -> B in Analysis
paulson <lp15@cam.ac.uk>
parents:
78202
diff
changeset
|
3441 |
mtopology_of_def) |
78202 | 3442 |
show "(mtopology_of m') closure_of f ` topspace X = topspace (mtopology_of m')" |
3443 |
by (simp add: Xeq m') |
|
3444 |
qed |
|
3445 |
qed |
|
3446 |
||
3447 |
||
3448 |
subsection\<open>Contractions\<close> |
|
3449 |
||
3450 |
lemma (in Metric_space) contraction_imp_unique_fixpoint: |
|
3451 |
assumes "f x = x" "f y = y" |
|
78248
740b23f1138a
EXPERIMENTAL replacement of f ` A <= B by f : A -> B in Analysis
paulson <lp15@cam.ac.uk>
parents:
78202
diff
changeset
|
3452 |
and "f \<in> M \<rightarrow> M" |
78202 | 3453 |
and "k < 1" |
3454 |
and "\<And>x y. \<lbrakk>x \<in> M; y \<in> M\<rbrakk> \<Longrightarrow> d (f x) (f y) \<le> k * d x y" |
|
3455 |
and "x \<in> M" "y \<in> M" |
|
3456 |
shows "x = y" |
|
78283 | 3457 |
by (smt (verit, ccfv_SIG) mdist_pos_less mult_le_cancel_right1 assms) |
78202 | 3458 |
|
3459 |
text \<open>Banach Fixed-Point Theorem (aka, Contraction Mapping Principle)\<close> |
|
3460 |
lemma (in Metric_space) Banach_fixedpoint_thm: |
|
78248
740b23f1138a
EXPERIMENTAL replacement of f ` A <= B by f : A -> B in Analysis
paulson <lp15@cam.ac.uk>
parents:
78202
diff
changeset
|
3461 |
assumes mcomplete and "M \<noteq> {}" and fim: "f \<in> M \<rightarrow> M" |
78202 | 3462 |
and "k < 1" |
3463 |
and con: "\<And>x y. \<lbrakk>x \<in> M; y \<in> M\<rbrakk> \<Longrightarrow> d (f x) (f y) \<le> k * d x y" |
|
3464 |
obtains x where "x \<in> M" "f x = x" |
|
3465 |
proof - |
|
3466 |
obtain a where "a \<in> M" |
|
3467 |
using \<open>M \<noteq> {}\<close> by blast |
|
3468 |
show thesis |
|
3469 |
proof (cases "\<forall>x \<in> M. f x = f a") |
|
3470 |
case True |
|
3471 |
then show ?thesis |
|
78248
740b23f1138a
EXPERIMENTAL replacement of f ` A <= B by f : A -> B in Analysis
paulson <lp15@cam.ac.uk>
parents:
78202
diff
changeset
|
3472 |
by (metis \<open>a \<in> M\<close> fim image_subset_iff image_subset_iff_funcset that) |
78202 | 3473 |
next |
3474 |
case False |
|
3475 |
then obtain b where "b \<in> M" and b: "f b \<noteq> f a" |
|
3476 |
by blast |
|
3477 |
have "k>0" |
|
3478 |
using Lipschitz_coefficient_pos [where f=f] |
|
3479 |
by (metis False \<open>a \<in> M\<close> con fim mdist_Self mspace_Self) |
|
3480 |
define \<sigma> where "\<sigma> \<equiv> \<lambda>n. (f^^n) a" |
|
3481 |
have f_iter: "\<sigma> n \<in> M" for n |
|
3482 |
unfolding \<sigma>_def by (induction n) (use \<open>a \<in> M\<close> fim in auto) |
|
3483 |
show ?thesis |
|
3484 |
proof (cases "f a = a") |
|
3485 |
case True |
|
3486 |
then show ?thesis |
|
3487 |
using \<open>a \<in> M\<close> that by blast |
|
3488 |
next |
|
3489 |
case False |
|
3490 |
have "MCauchy \<sigma>" |
|
3491 |
proof - |
|
3492 |
show ?thesis |
|
3493 |
unfolding MCauchy_def |
|
3494 |
proof (intro conjI strip) |
|
3495 |
show "range \<sigma> \<subseteq> M" |
|
3496 |
using f_iter by blast |
|
3497 |
fix \<epsilon>::real |
|
3498 |
assume "\<epsilon>>0" |
|
3499 |
with \<open>k < 1\<close> \<open>f a \<noteq> a\<close> \<open>a \<in> M\<close> fim have gt0: "((1 - k) * \<epsilon>) / d a (f a) > 0" |
|
78248
740b23f1138a
EXPERIMENTAL replacement of f ` A <= B by f : A -> B in Analysis
paulson <lp15@cam.ac.uk>
parents:
78202
diff
changeset
|
3500 |
by (fastforce simp: divide_simps Pi_iff) |
78202 | 3501 |
obtain N where "k^N < ((1-k) * \<epsilon>) / d a (f a)" |
3502 |
using real_arch_pow_inv [OF gt0 \<open>k < 1\<close>] by blast |
|
3503 |
then have N: "\<And>n. n \<ge> N \<Longrightarrow> k^n < ((1-k) * \<epsilon>) / d a (f a)" |
|
3504 |
by (smt (verit) \<open>0 < k\<close> assms(4) power_decreasing) |
|
3505 |
have "\<forall>n n'. n<n' \<longrightarrow> N \<le> n \<longrightarrow> N \<le> n' \<longrightarrow> d (\<sigma> n) (\<sigma> n') < \<epsilon>" |
|
3506 |
proof (intro exI strip) |
|
3507 |
fix n n' |
|
3508 |
assume "n<n'" "N \<le> n" "N \<le> n'" |
|
3509 |
have "d (\<sigma> n) (\<sigma> n') \<le> (\<Sum>i=n..<n'. d (\<sigma> i) (\<sigma> (Suc i)))" |
|
3510 |
proof - |
|
3511 |
have "n < m \<Longrightarrow> d (\<sigma> n) (\<sigma> m) \<le> (\<Sum>i=n..<m. d (\<sigma> i) (\<sigma> (Suc i)))" for m |
|
3512 |
proof (induction m) |
|
3513 |
case 0 |
|
3514 |
then show ?case |
|
3515 |
by simp |
|
3516 |
next |
|
3517 |
case (Suc m) |
|
3518 |
then consider "n<m" | "m=n" |
|
3519 |
by linarith |
|
3520 |
then show ?case |
|
3521 |
proof cases |
|
3522 |
case 1 |
|
3523 |
have "d (\<sigma> n) (\<sigma> (Suc m)) \<le> d (\<sigma> n) (\<sigma> m) + d (\<sigma> m) (\<sigma> (Suc m))" |
|
3524 |
by (simp add: f_iter triangle) |
|
3525 |
also have "\<dots> \<le> (\<Sum>i=n..<m. d (\<sigma> i) (\<sigma> (Suc i))) + d (\<sigma> m) (\<sigma> (Suc m))" |
|
3526 |
using Suc 1 by linarith |
|
3527 |
also have "\<dots> = (\<Sum>i = n..<Suc m. d (\<sigma> i) (\<sigma> (Suc i)))" |
|
3528 |
using "1" by force |
|
3529 |
finally show ?thesis . |
|
3530 |
qed auto |
|
3531 |
qed |
|
3532 |
with \<open>n < n'\<close> show ?thesis by blast |
|
3533 |
qed |
|
3534 |
also have "\<dots> \<le> (\<Sum>i=n..<n'. d a (f a) * k^i)" |
|
3535 |
proof (rule sum_mono) |
|
3536 |
fix i |
|
3537 |
assume "i \<in> {n..<n'}" |
|
3538 |
show "d (\<sigma> i) (\<sigma> (Suc i)) \<le> d a (f a) * k ^ i" |
|
3539 |
proof (induction i) |
|
3540 |
case 0 |
|
3541 |
then show ?case |
|
3542 |
by (auto simp: \<sigma>_def) |
|
3543 |
next |
|
3544 |
case (Suc i) |
|
3545 |
have "d (\<sigma> (Suc i)) (\<sigma> (Suc (Suc i))) \<le> k * d (\<sigma> i) (\<sigma> (Suc i))" |
|
3546 |
using con \<sigma>_def f_iter fim by fastforce |
|
3547 |
also have "\<dots> \<le> d a (f a) * k ^ Suc i" |
|
3548 |
using Suc \<open>0 < k\<close> by auto |
|
3549 |
finally show ?case . |
|
3550 |
qed |
|
3551 |
qed |
|
3552 |
also have "\<dots> = d a (f a) * (\<Sum>i=n..<n'. k^i)" |
|
3553 |
by (simp add: sum_distrib_left) |
|
3554 |
also have "\<dots> = d a (f a) * (\<Sum>i=0..<n'-n. k^(i+n))" |
|
3555 |
using sum.shift_bounds_nat_ivl [of "power k" 0 n "n'-n"] \<open>n < n'\<close> by simp |
|
3556 |
also have "\<dots> = d a (f a) * k^n * (\<Sum>i<n'-n. k^i)" |
|
3557 |
by (simp add: power_add lessThan_atLeast0 flip: sum_distrib_right) |
|
3558 |
also have "\<dots> = d a (f a) * (k ^ n - k ^ n') / (1 - k)" |
|
3559 |
using \<open>k < 1\<close> \<open>n < n'\<close> apply (simp add: sum_gp_strict) |
|
3560 |
by (simp add: algebra_simps flip: power_add) |
|
3561 |
also have "\<dots> < \<epsilon>" |
|
3562 |
using N \<open>k < 1\<close> \<open>0 < \<epsilon>\<close> \<open>0 < k\<close> \<open>N \<le> n\<close> |
|
3563 |
apply (simp add: field_simps) |
|
3564 |
by (smt (verit) nonneg pos_less_divide_eq zero_less_divide_iff zero_less_power) |
|
3565 |
finally show "d (\<sigma> n) (\<sigma> n') < \<epsilon>" . |
|
3566 |
qed |
|
3567 |
then show "\<exists>N. \<forall>n n'. N \<le> n \<longrightarrow> N \<le> n' \<longrightarrow> d (\<sigma> n) (\<sigma> n') < \<epsilon>" |
|
3568 |
by (metis \<open>0 < \<epsilon>\<close> commute f_iter linorder_not_le local.mdist_zero nat_less_le) |
|
3569 |
qed |
|
3570 |
qed |
|
3571 |
then obtain l where l: "limitin mtopology \<sigma> l sequentially" |
|
3572 |
using \<open>mcomplete\<close> mcomplete_def by blast |
|
3573 |
show ?thesis |
|
3574 |
proof |
|
3575 |
show "l \<in> M" |
|
3576 |
using l limitin_mspace by blast |
|
3577 |
show "f l = l" |
|
3578 |
proof (rule limitin_metric_unique) |
|
3579 |
have "limitin mtopology (f \<circ> \<sigma>) (f l) sequentially" |
|
3580 |
proof (rule continuous_map_limit) |
|
3581 |
have "Lipschitz_continuous_map Self Self f" |
|
3582 |
using con by (auto simp: Lipschitz_continuous_map_def fim) |
|
3583 |
then show "continuous_map mtopology mtopology f" |
|
3584 |
using Lipschitz_continuous_imp_continuous_map Self_def by force |
|
3585 |
qed (use l in auto) |
|
3586 |
moreover have "(f \<circ> \<sigma>) = (\<lambda>i. \<sigma>(i+1))" |
|
3587 |
by (auto simp: \<sigma>_def) |
|
3588 |
ultimately show "limitin mtopology (\<lambda>n. (f^^n)a) (f l) sequentially" |
|
3589 |
using limitin_sequentially_offset_rev [of mtopology \<sigma> 1] |
|
3590 |
by (simp add: \<sigma>_def) |
|
3591 |
qed (use l in \<open>auto simp: \<sigma>_def\<close>) |
|
3592 |
qed |
|
3593 |
qed |
|
3594 |
qed |
|
3595 |
qed |
|
3596 |
||
3597 |
||
3598 |
subsection\<open> The Baire Category Theorem\<close> |
|
3599 |
||
3600 |
text \<open>Possibly relevant to the theorem "Baire" in Elementary Normed Spaces\<close> |
|
3601 |
||
3602 |
lemma (in Metric_space) metric_Baire_category: |
|
3603 |
assumes "mcomplete" "countable \<G>" |
|
3604 |
and "\<And>T. T \<in> \<G> \<Longrightarrow> openin mtopology T \<and> mtopology closure_of T = M" |
|
3605 |
shows "mtopology closure_of \<Inter>\<G> = M" |
|
3606 |
proof (cases "\<G>={}") |
|
3607 |
case False |
|
3608 |
then obtain U :: "nat \<Rightarrow> 'a set" where U: "range U = \<G>" |
|
3609 |
by (metis \<open>countable \<G>\<close> uncountable_def) |
|
3610 |
with assms have u_open: "\<And>n. openin mtopology (U n)" and u_dense: "\<And>n. mtopology closure_of (U n) = M" |
|
3611 |
by auto |
|
3612 |
have "\<Inter>(range U) \<inter> W \<noteq> {}" if W: "openin mtopology W" "W \<noteq> {}" for W |
|
3613 |
proof - |
|
3614 |
have "W \<subseteq> M" |
|
3615 |
using openin_mtopology W by blast |
|
3616 |
have "\<exists>r' x'. 0 < r' \<and> r' < r/2 \<and> x' \<in> M \<and> mcball x' r' \<subseteq> mball x r \<inter> U n" |
|
3617 |
if "r>0" "x \<in> M" for x r n |
|
3618 |
proof - |
|
3619 |
obtain z where z: "z \<in> U n \<inter> mball x r" |
|
3620 |
using u_dense [of n] \<open>r>0\<close> \<open>x \<in> M\<close> |
|
3621 |
by (metis dense_intersects_open centre_in_mball_iff empty_iff openin_mball topspace_mtopology equals0I) |
|
3622 |
then have "z \<in> M" by auto |
|
3623 |
have "openin mtopology (U n \<inter> mball x r)" |
|
3624 |
by (simp add: openin_Int u_open) |
|
3625 |
with \<open>z \<in> M\<close> z obtain e where "e>0" and e: "mcball z e \<subseteq> U n \<inter> mball x r" |
|
3626 |
by (meson openin_mtopology_mcball) |
|
3627 |
define r' where "r' \<equiv> min e (r/4)" |
|
3628 |
show ?thesis |
|
3629 |
proof (intro exI conjI) |
|
3630 |
show "0 < r'" "r' < r / 2" "z \<in> M" |
|
3631 |
using \<open>e>0\<close> \<open>r>0\<close> \<open>z \<in> M\<close> by (auto simp: r'_def) |
|
3632 |
show "mcball z r' \<subseteq> mball x r \<inter> U n" |
|
3633 |
using Metric_space.mcball_subset_concentric e r'_def by auto |
|
3634 |
qed |
|
3635 |
qed |
|
3636 |
then obtain nextr nextx |
|
3637 |
where nextr: "\<And>r x n. \<lbrakk>r>0; x\<in>M\<rbrakk> \<Longrightarrow> 0 < nextr r x n \<and> nextr r x n < r/2" |
|
3638 |
and nextx: "\<And>r x n. \<lbrakk>r>0; x\<in>M\<rbrakk> \<Longrightarrow> nextx r x n \<in> M" |
|
3639 |
and nextsub: "\<And>r x n. \<lbrakk>r>0; x\<in>M\<rbrakk> \<Longrightarrow> mcball (nextx r x n) (nextr r x n) \<subseteq> mball x r \<inter> U n" |
|
3640 |
by metis |
|
3641 |
obtain x0 where x0: "x0 \<in> U 0 \<inter> W" |
|
3642 |
by (metis W dense_intersects_open topspace_mtopology all_not_in_conv u_dense) |
|
3643 |
then have "x0 \<in> M" |
|
3644 |
using \<open>W \<subseteq> M\<close> by fastforce |
|
3645 |
obtain r0 where "0 < r0" "r0 < 1" and sub: "mcball x0 r0 \<subseteq> U 0 \<inter> W" |
|
3646 |
proof - |
|
3647 |
have "openin mtopology (U 0 \<inter> W)" |
|
3648 |
using W u_open by blast |
|
3649 |
then obtain r where "r>0" and r: "mball x0 r \<subseteq> U 0" "mball x0 r \<subseteq> W" |
|
3650 |
by (meson Int_subset_iff openin_mtopology x0) |
|
3651 |
define r0 where "r0 \<equiv> (min r 1) / 2" |
|
3652 |
show thesis |
|
3653 |
proof |
|
3654 |
show "0 < r0" "r0 < 1" |
|
3655 |
using \<open>r>0\<close> by (auto simp: r0_def) |
|
3656 |
show "mcball x0 r0 \<subseteq> U 0 \<inter> W" |
|
3657 |
using r \<open>0 < r0\<close> r0_def by auto |
|
3658 |
qed |
|
3659 |
qed |
|
3660 |
define b where "b \<equiv> rec_nat (x0,r0) (\<lambda>n (x,r). (nextx r x n, nextr r x n))" |
|
3661 |
have b0[simp]: "b 0 = (x0,r0)" |
|
3662 |
by (simp add: b_def) |
|
3663 |
have bSuc[simp]: "b (Suc n) = (let (x,r) = b n in (nextx r x n, nextr r x n))" for n |
|
3664 |
by (simp add: b_def) |
|
3665 |
define xf where "xf \<equiv> fst \<circ> b" |
|
3666 |
define rf where "rf \<equiv> snd \<circ> b" |
|
3667 |
have rfxf: "0 < rf n \<and> xf n \<in> M" for n |
|
3668 |
proof (induction n) |
|
3669 |
case 0 |
|
3670 |
with \<open>0 < r0\<close> \<open>x0 \<in> M\<close> show ?case |
|
3671 |
by (auto simp: rf_def xf_def) |
|
3672 |
next |
|
3673 |
case (Suc n) |
|
3674 |
then show ?case |
|
3675 |
by (auto simp: rf_def xf_def case_prod_unfold nextr nextx Let_def) |
|
3676 |
qed |
|
3677 |
have mcball_sub: "mcball (xf (Suc n)) (rf (Suc n)) \<subseteq> mball (xf n) (rf n) \<inter> U n" for n |
|
3678 |
using rfxf nextsub by (auto simp: xf_def rf_def case_prod_unfold Let_def) |
|
3679 |
have half: "rf (Suc n) < rf n / 2" for n |
|
3680 |
using rfxf nextr by (auto simp: xf_def rf_def case_prod_unfold Let_def) |
|
3681 |
then have "decseq rf" |
|
3682 |
using rfxf by (smt (verit, ccfv_threshold) decseq_SucI field_sum_of_halves) |
|
3683 |
have nested: "mball (xf n) (rf n) \<subseteq> mball (xf m) (rf m)" if "m \<le> n" for m n |
|
3684 |
using that |
|
3685 |
proof (induction n) |
|
3686 |
case (Suc n) |
|
3687 |
then show ?case |
|
3688 |
by (metis mcball_sub order.trans inf.boundedE le_Suc_eq mball_subset_mcball order.refl) |
|
3689 |
qed auto |
|
3690 |
have "MCauchy xf" |
|
3691 |
unfolding MCauchy_def |
|
3692 |
proof (intro conjI strip) |
|
3693 |
show "range xf \<subseteq> M" |
|
3694 |
using rfxf by blast |
|
3695 |
fix \<epsilon> :: real |
|
3696 |
assume "\<epsilon>>0" |
|
3697 |
then obtain N where N: "inverse (2^N) < \<epsilon>" |
|
3698 |
using real_arch_pow_inv by (force simp flip: power_inverse) |
|
3699 |
have "d (xf n) (xf n') < \<epsilon>" if "n \<le> n'" "N \<le> n" "N \<le> n'" for n n' |
|
3700 |
proof - |
|
3701 |
have *: "rf n < inverse (2 ^ n)" for n |
|
3702 |
proof (induction n) |
|
3703 |
case 0 |
|
3704 |
then show ?case |
|
3705 |
by (simp add: \<open>r0 < 1\<close> rf_def) |
|
3706 |
next |
|
3707 |
case (Suc n) |
|
3708 |
with half show ?case |
|
3709 |
by simp (smt (verit)) |
|
3710 |
qed |
|
3711 |
have "rf n \<le> rf N" |
|
3712 |
using \<open>decseq rf\<close> \<open>N \<le> n\<close> by (simp add: decseqD) |
|
3713 |
moreover |
|
3714 |
have "xf n' \<in> mball (xf n) (rf n)" |
|
3715 |
using nested rfxf \<open>n \<le> n'\<close> by blast |
|
3716 |
ultimately have "d (xf n) (xf n') < rf N" |
|
3717 |
by auto |
|
3718 |
also have "\<dots> < \<epsilon>" |
|
3719 |
using "*" N order.strict_trans by blast |
|
3720 |
finally show ?thesis . |
|
3721 |
qed |
|
3722 |
then show "\<exists>N. \<forall>n n'. N \<le> n \<longrightarrow> N \<le> n' \<longrightarrow> d (xf n) (xf n') < \<epsilon>" |
|
3723 |
by (metis commute linorder_le_cases) |
|
3724 |
qed |
|
3725 |
then obtain l where l: "limitin mtopology xf l sequentially" |
|
3726 |
using \<open>mcomplete\<close> mcomplete_alt by blast |
|
3727 |
have l_in: "l \<in> mcball (xf n) (rf n)" for n |
|
3728 |
proof - |
|
3729 |
have "\<forall>\<^sub>F m in sequentially. xf m \<in> mcball (xf n) (rf n)" |
|
3730 |
unfolding eventually_sequentially |
|
3731 |
by (meson nested rfxf centre_in_mball_iff mball_subset_mcball subset_iff) |
|
3732 |
with l limitin_closedin show ?thesis |
|
3733 |
by (metis closedin_mcball trivial_limit_sequentially) |
|
3734 |
qed |
|
3735 |
then have "\<And>n. l \<in> U n" |
|
3736 |
using mcball_sub by blast |
|
3737 |
moreover have "l \<in> W" |
|
3738 |
using l_in[of 0] sub by (auto simp: xf_def rf_def) |
|
3739 |
ultimately show ?thesis by auto |
|
3740 |
qed |
|
3741 |
with U show ?thesis |
|
3742 |
by (metis dense_intersects_open topspace_mtopology) |
|
3743 |
qed auto |
|
3744 |
||
3745 |
||
3746 |
||
3747 |
lemma (in Metric_space) metric_Baire_category_alt: |
|
3748 |
assumes "mcomplete" "countable \<G>" |
|
3749 |
and empty: "\<And>T. T \<in> \<G> \<Longrightarrow> closedin mtopology T \<and> mtopology interior_of T = {}" |
|
3750 |
shows "mtopology interior_of \<Union>\<G> = {}" |
|
3751 |
proof - |
|
3752 |
have *: "mtopology closure_of \<Inter>((-)M ` \<G>) = M" |
|
3753 |
proof (intro metric_Baire_category conjI \<open>mcomplete\<close>) |
|
3754 |
show "countable ((-) M ` \<G>)" |
|
3755 |
using \<open>countable \<G>\<close> by blast |
|
3756 |
fix T |
|
3757 |
assume "T \<in> (-) M ` \<G>" |
|
3758 |
then obtain U where U: "U \<in> \<G>" "T = M-U" "U \<subseteq> M" |
|
3759 |
using empty metric_closedin_iff_sequentially_closed by force |
|
3760 |
with empty show "openin mtopology T" by blast |
|
3761 |
show "mtopology closure_of T = M" |
|
3762 |
using U by (simp add: closure_of_interior_of double_diff empty) |
|
3763 |
qed |
|
3764 |
with closure_of_eq show ?thesis |
|
3765 |
by (fastforce simp: interior_of_closure_of split: if_split_asm) |
|
3766 |
qed |
|
3767 |
||
3768 |
||
3769 |
text \<open>Since all locally compact Hausdorff spaces are regular, the disjunction in the HOL Light version is redundant.\<close> |
|
3770 |
lemma Baire_category_aux: |
|
3771 |
assumes "locally_compact_space X" "regular_space X" |
|
3772 |
and "countable \<G>" |
|
3773 |
and empty: "\<And>G. G \<in> \<G> \<Longrightarrow> closedin X G \<and> X interior_of G = {}" |
|
3774 |
shows "X interior_of \<Union>\<G> = {}" |
|
3775 |
proof (cases "\<G> = {}") |
|
3776 |
case True |
|
3777 |
then show ?thesis |
|
3778 |
by simp |
|
3779 |
next |
|
3780 |
case False |
|
3781 |
then obtain T :: "nat \<Rightarrow> 'a set" where T: "\<G> = range T" |
|
3782 |
by (metis \<open>countable \<G>\<close> uncountable_def) |
|
3783 |
with empty have Tempty: "\<And>n. X interior_of (T n) = {}" |
|
3784 |
by auto |
|
3785 |
show ?thesis |
|
3786 |
proof (clarsimp simp: T interior_of_def) |
|
3787 |
fix z U |
|
3788 |
assume "z \<in> U" and opeA: "openin X U" and Asub: "U \<subseteq> \<Union> (range T)" |
|
3789 |
with openin_subset have "z \<in> topspace X" |
|
3790 |
by blast |
|
3791 |
have "neighbourhood_base_of (\<lambda>C. compactin X C \<and> closedin X C) X" |
|
3792 |
using assms locally_compact_regular_space_neighbourhood_base by auto |
|
3793 |
then obtain V K where "openin X V" "compactin X K" "closedin X K" "z \<in> V" "V \<subseteq> K" "K \<subseteq> U" |
|
3794 |
by (metis (no_types, lifting) \<open>z \<in> U\<close> neighbourhood_base_of opeA) |
|
3795 |
have nb_closedin: "neighbourhood_base_of (closedin X) X" |
|
3796 |
using \<open>regular_space X\<close> neighbourhood_base_of_closedin by auto |
|
3797 |
have "\<exists>\<Phi>. \<forall>n. (\<Phi> n \<subseteq> K \<and> closedin X (\<Phi> n) \<and> X interior_of \<Phi> n \<noteq> {} \<and> disjnt (\<Phi> n) (T n)) \<and> |
|
3798 |
\<Phi> (Suc n) \<subseteq> \<Phi> n" |
|
3799 |
proof (rule dependent_nat_choice) |
|
3800 |
show "\<exists>x\<subseteq>K. closedin X x \<and> X interior_of x \<noteq> {} \<and> disjnt x (T 0)" |
|
3801 |
proof - |
|
3802 |
have False if "V \<subseteq> T 0" |
|
3803 |
using Tempty \<open>openin X V\<close> \<open>z \<in> V\<close> interior_of_maximal that by fastforce |
|
3804 |
then obtain x where "openin X (V - T 0) \<and> x \<in> V - T 0" |
|
3805 |
using T \<open>openin X V\<close> empty by blast |
|
3806 |
with nb_closedin |
|
3807 |
obtain N C where "openin X N" "closedin X C" "x \<in> N" "N \<subseteq> C" "C \<subseteq> V - T 0" |
|
3808 |
unfolding neighbourhood_base_of by metis |
|
3809 |
show ?thesis |
|
3810 |
proof (intro exI conjI) |
|
3811 |
show "C \<subseteq> K" |
|
3812 |
using \<open>C \<subseteq> V - T 0\<close> \<open>V \<subseteq> K\<close> by auto |
|
3813 |
show "X interior_of C \<noteq> {}" |
|
3814 |
by (metis \<open>N \<subseteq> C\<close> \<open>openin X N\<close> \<open>x \<in> N\<close> empty_iff interior_of_eq_empty) |
|
3815 |
show "disjnt C (T 0)" |
|
3816 |
using \<open>C \<subseteq> V - T 0\<close> disjnt_iff by fastforce |
|
3817 |
qed (use \<open>closedin X C\<close> in auto) |
|
3818 |
qed |
|
3819 |
show "\<exists>L. (L \<subseteq> K \<and> closedin X L \<and> X interior_of L \<noteq> {} \<and> disjnt L (T (Suc n))) \<and> L \<subseteq> C" |
|
3820 |
if \<section>: "C \<subseteq> K \<and> closedin X C \<and> X interior_of C \<noteq> {} \<and> disjnt C (T n)" |
|
3821 |
for C n |
|
3822 |
proof - |
|
3823 |
have False if "X interior_of C \<subseteq> T (Suc n)" |
|
3824 |
by (metis Tempty interior_of_eq_empty \<section> openin_interior_of that) |
|
3825 |
then obtain x where "openin X (X interior_of C - T (Suc n)) \<and> x \<in> X interior_of C - T (Suc n)" |
|
3826 |
using T empty by fastforce |
|
3827 |
with nb_closedin |
|
3828 |
obtain N D where "openin X N" "closedin X D" "x \<in> N" "N \<subseteq> D" and D: "D \<subseteq> X interior_of C - T(Suc n)" |
|
3829 |
unfolding neighbourhood_base_of by metis |
|
3830 |
show ?thesis |
|
3831 |
proof (intro conjI exI) |
|
3832 |
show "D \<subseteq> K" |
|
3833 |
using D interior_of_subset \<section> by fastforce |
|
3834 |
show "X interior_of D \<noteq> {}" |
|
3835 |
by (metis \<open>N \<subseteq> D\<close> \<open>openin X N\<close> \<open>x \<in> N\<close> empty_iff interior_of_eq_empty) |
|
3836 |
show "disjnt D (T (Suc n))" |
|
3837 |
using D disjnt_iff by fastforce |
|
3838 |
show "D \<subseteq> C" |
|
3839 |
using interior_of_subset [of X C] D by blast |
|
3840 |
qed (use \<open>closedin X D\<close> in auto) |
|
3841 |
qed |
|
3842 |
qed |
|
3843 |
then obtain \<Phi> where \<Phi>: "\<And>n. \<Phi> n \<subseteq> K \<and> closedin X (\<Phi> n) \<and> X interior_of \<Phi> n \<noteq> {} \<and> disjnt (\<Phi> n) (T n)" |
|
3844 |
and "\<And>n. \<Phi> (Suc n) \<subseteq> \<Phi> n" by metis |
|
3845 |
then have "decseq \<Phi>" |
|
3846 |
by (simp add: decseq_SucI) |
|
3847 |
moreover have "\<And>n. \<Phi> n \<noteq> {}" |
|
3848 |
by (metis \<Phi> bot.extremum_uniqueI interior_of_subset) |
|
3849 |
ultimately have "\<Inter>(range \<Phi>) \<noteq> {}" |
|
3850 |
by (metis \<Phi> compact_space_imp_nest \<open>compactin X K\<close> compactin_subspace closedin_subset_topspace) |
|
3851 |
moreover have "U \<subseteq> {y. \<exists>x. y \<in> T x}" |
|
3852 |
using Asub by auto |
|
3853 |
with T have "{a. \<forall>n. a \<in> \<Phi> n} \<subseteq> {}" |
|
3854 |
by (smt (verit) Asub \<Phi> Collect_empty_eq UN_iff \<open>K \<subseteq> U\<close> disjnt_iff subset_iff) |
|
3855 |
ultimately show False |
|
3856 |
by blast |
|
3857 |
qed |
|
3858 |
qed |
|
3859 |
||
3860 |
||
3861 |
lemma Baire_category_alt: |
|
3862 |
assumes "completely_metrizable_space X \<or> locally_compact_space X \<and> regular_space X" |
|
3863 |
and "countable \<G>" |
|
3864 |
and "\<And>T. T \<in> \<G> \<Longrightarrow> closedin X T \<and> X interior_of T = {}" |
|
3865 |
shows "X interior_of \<Union>\<G> = {}" |
|
3866 |
using Baire_category_aux [of X \<G>] Metric_space.metric_Baire_category_alt |
|
3867 |
by (metis assms completely_metrizable_space_def) |
|
3868 |
||
3869 |
||
3870 |
lemma Baire_category: |
|
3871 |
assumes "completely_metrizable_space X \<or> locally_compact_space X \<and> regular_space X" |
|
3872 |
and "countable \<G>" |
|
3873 |
and top: "\<And>T. T \<in> \<G> \<Longrightarrow> openin X T \<and> X closure_of T = topspace X" |
|
3874 |
shows "X closure_of \<Inter>\<G> = topspace X" |
|
3875 |
proof (cases "\<G>={}") |
|
3876 |
case False |
|
3877 |
have *: "X interior_of \<Union>((-)(topspace X) ` \<G>) = {}" |
|
3878 |
proof (intro Baire_category_alt conjI assms) |
|
3879 |
show "countable ((-) (topspace X) ` \<G>)" |
|
3880 |
using assms by blast |
|
3881 |
fix T |
|
3882 |
assume "T \<in> (-) (topspace X) ` \<G>" |
|
3883 |
then obtain U where U: "U \<in> \<G>" "T = (topspace X) - U" "U \<subseteq> (topspace X)" |
|
3884 |
by (meson top image_iff openin_subset) |
|
3885 |
then show "closedin X T" |
|
3886 |
by (simp add: closedin_diff top) |
|
3887 |
show "X interior_of T = {}" |
|
3888 |
using U top by (simp add: interior_of_closure_of double_diff) |
|
3889 |
qed |
|
3890 |
then show ?thesis |
|
3891 |
by (simp add: closure_of_eq_topspace interior_of_complement) |
|
3892 |
qed auto |
|
3893 |
||
3894 |
||
3895 |
subsection\<open>Sierpinski-Hausdorff type results about countable closed unions\<close> |
|
3896 |
||
3897 |
lemma locally_connected_not_countable_closed_union: |
|
3898 |
assumes "topspace X \<noteq> {}" and csX: "connected_space X" |
|
3899 |
and lcX: "locally_connected_space X" |
|
3900 |
and X: "completely_metrizable_space X \<or> locally_compact_space X \<and> Hausdorff_space X" |
|
3901 |
and "countable \<U>" and pwU: "pairwise disjnt \<U>" |
|
3902 |
and clo: "\<And>C. C \<in> \<U> \<Longrightarrow> closedin X C \<and> C \<noteq> {}" |
|
3903 |
and UU_eq: "\<Union>\<U> = topspace X" |
|
3904 |
shows "\<U> = {topspace X}" |
|
3905 |
proof - |
|
3906 |
define \<V> where "\<V> \<equiv> (frontier_of) X ` \<U>" |
|
3907 |
define B where "B \<equiv> \<Union>\<V>" |
|
3908 |
then have Bsub: "B \<subseteq> topspace X" |
|
3909 |
by (simp add: Sup_le_iff \<V>_def closedin_frontier_of closedin_subset) |
|
3910 |
have allsub: "A \<subseteq> topspace X" if "A \<in> \<U>" for A |
|
3911 |
by (meson clo closedin_def that) |
|
3912 |
show ?thesis |
|
3913 |
proof (rule ccontr) |
|
3914 |
assume "\<U> \<noteq> {topspace X}" |
|
3915 |
with assms have "\<exists>A\<in>\<U>. \<not> (closedin X A \<and> openin X A)" |
|
3916 |
by (metis Union_empty connected_space_clopen_in singletonI subsetI subset_singleton_iff) |
|
3917 |
then have "B \<noteq> {}" |
|
3918 |
by (auto simp: B_def \<V>_def frontier_of_eq_empty allsub) |
|
3919 |
moreover |
|
3920 |
have "subtopology X B interior_of B = B" |
|
3921 |
by (simp add: Bsub interior_of_openin openin_subtopology_refl) |
|
3922 |
ultimately have int_B_nonempty: "subtopology X B interior_of B \<noteq> {}" |
|
3923 |
by auto |
|
3924 |
have "subtopology X B interior_of \<Union>\<V> = {}" |
|
3925 |
proof (intro Baire_category_alt conjI) |
|
3926 |
have "\<Union>\<U> \<subseteq> B \<union> \<Union>((interior_of) X ` \<U>)" |
|
3927 |
using clo closure_of_closedin by (fastforce simp: B_def \<V>_def frontier_of_def) |
|
3928 |
moreover have "B \<union> \<Union>((interior_of) X ` \<U>) \<subseteq> \<Union>\<U>" |
|
3929 |
using allsub clo frontier_of_subset_eq interior_of_subset by (fastforce simp: B_def \<V>_def ) |
|
3930 |
moreover have "disjnt B (\<Union>((interior_of) X ` \<U>))" |
|
3931 |
using pwU |
|
3932 |
apply (clarsimp simp: B_def \<V>_def frontier_of_def pairwise_def disjnt_iff) |
|
3933 |
by (metis clo closure_of_eq interior_of_subset subsetD) |
|
3934 |
ultimately have "B = topspace X - \<Union> ((interior_of) X ` \<U>)" |
|
3935 |
by (auto simp: UU_eq disjnt_iff) |
|
3936 |
then have "closedin X B" |
|
3937 |
by fastforce |
|
3938 |
with X show "completely_metrizable_space (subtopology X B) \<or> locally_compact_space (subtopology X B) \<and> regular_space (subtopology X B)" |
|
3939 |
by (metis completely_metrizable_space_closedin locally_compact_Hausdorff_or_regular |
|
3940 |
locally_compact_space_closed_subset regular_space_subtopology) |
|
3941 |
show "countable \<V>" |
|
3942 |
by (simp add: \<V>_def \<open>countable \<U>\<close>) |
|
3943 |
fix V |
|
3944 |
assume "V \<in> \<V>" |
|
3945 |
then obtain S where S: "S \<in> \<U>" "V = X frontier_of S" |
|
3946 |
by (auto simp: \<V>_def) |
|
3947 |
show "closedin (subtopology X B) V" |
|
3948 |
by (metis B_def Sup_upper \<V>_def \<open>V \<in> \<V>\<close> closedin_frontier_of closedin_subset_topspace image_iff) |
|
3949 |
have "subtopology X B interior_of (X frontier_of S) = {}" |
|
3950 |
proof (clarsimp simp: interior_of_def openin_subtopology_alt) |
|
3951 |
fix a U |
|
3952 |
assume "a \<in> B" "a \<in> U" and opeU: "openin X U" and BUsub: "B \<inter> U \<subseteq> X frontier_of S" |
|
3953 |
then have "a \<in> S" |
|
3954 |
by (meson IntI \<open>S \<in> \<U>\<close> clo frontier_of_subset_closedin subsetD) |
|
3955 |
then obtain W C where "openin X W" "connectedin X C" "a \<in> W" "W \<subseteq> C" "C \<subseteq> U" |
|
3956 |
by (metis \<open>a \<in> U\<close> lcX locally_connected_space opeU) |
|
3957 |
have "W \<inter> X frontier_of S \<noteq> {}" |
|
3958 |
using \<open>B \<inter> U \<subseteq> X frontier_of S\<close> \<open>a \<in> B\<close> \<open>a \<in> U\<close> \<open>a \<in> W\<close> by auto |
|
3959 |
with frontier_of_openin_straddle_Int |
|
3960 |
obtain "W \<inter> S \<noteq> {}" "W - S \<noteq> {}" "W \<subseteq> topspace X" |
|
3961 |
using \<open>openin X W\<close> by (metis openin_subset) |
|
3962 |
then obtain b where "b \<in> topspace X" "b \<in> W-S" |
|
3963 |
by blast |
|
3964 |
with UU_eq obtain T where "T \<in> \<U>" "T \<noteq> S" "W \<inter> T \<noteq> {}" |
|
3965 |
by auto |
|
3966 |
then have "disjnt S T" |
|
3967 |
by (metis \<open>S \<in> \<U>\<close> pairwise_def pwU) |
|
3968 |
then have "C - T \<noteq> {}" |
|
3969 |
by (meson Diff_eq_empty_iff \<open>W \<subseteq> C\<close> \<open>a \<in> S\<close> \<open>a \<in> W\<close> disjnt_iff subsetD) |
|
3970 |
then have "C \<inter> X frontier_of T \<noteq> {}" |
|
3971 |
using \<open>W \<inter> T \<noteq> {}\<close> \<open>W \<subseteq> C\<close> \<open>connectedin X C\<close> connectedin_Int_frontier_of by blast |
|
3972 |
moreover have "C \<inter> X frontier_of T = {}" |
|
3973 |
proof - |
|
3974 |
have "X frontier_of S \<subseteq> S" "X frontier_of T \<subseteq> T" |
|
3975 |
using frontier_of_subset_closedin \<open>S \<in> \<U>\<close> \<open>T \<in> \<U>\<close> clo by blast+ |
|
3976 |
moreover have "X frontier_of T \<union> B = B" |
|
3977 |
using B_def \<V>_def \<open>T \<in> \<U>\<close> by blast |
|
3978 |
ultimately show ?thesis |
|
3979 |
using BUsub \<open>C \<subseteq> U\<close> \<open>disjnt S T\<close> unfolding disjnt_def by blast |
|
3980 |
qed |
|
3981 |
ultimately show False |
|
3982 |
by simp |
|
3983 |
qed |
|
3984 |
with S show "subtopology X B interior_of V = {}" |
|
3985 |
by meson |
|
3986 |
qed |
|
3987 |
then show False |
|
3988 |
using B_def int_B_nonempty by blast |
|
3989 |
qed |
|
3990 |
qed |
|
3991 |
||
3992 |
lemma real_Sierpinski_lemma: |
|
3993 |
fixes a b::real |
|
3994 |
assumes "a \<le> b" |
|
3995 |
and "countable \<U>" and pwU: "pairwise disjnt \<U>" |
|
3996 |
and clo: "\<And>C. C \<in> \<U> \<Longrightarrow> closed C \<and> C \<noteq> {}" |
|
3997 |
and "\<Union>\<U> = {a..b}" |
|
3998 |
shows "\<U> = {{a..b}}" |
|
3999 |
proof - |
|
4000 |
have "locally_connected_space (top_of_set {a..b})" |
|
4001 |
by (simp add: locally_connected_real_interval) |
|
4002 |
moreover |
|
4003 |
have "completely_metrizable_space (top_of_set {a..b})" |
|
4004 |
by (metis box_real(2) completely_metrizable_space_cbox) |
|
4005 |
ultimately |
|
4006 |
show ?thesis |
|
4007 |
using locally_connected_not_countable_closed_union [of "subtopology euclidean {a..b}"] assms |
|
4008 |
apply (simp add: closedin_subtopology) |
|
4009 |
by (metis Union_upper inf.orderE) |
|
4010 |
qed |
|
4011 |
||
4012 |
||
4013 |
subsection\<open>The Tychonoff embedding\<close> |
|
4014 |
||
4015 |
lemma completely_regular_space_cube_embedding_explicit: |
|
4016 |
assumes "completely_regular_space X" "Hausdorff_space X" |
|
4017 |
shows "embedding_map X |
|
4018 |
(product_topology (\<lambda>f. top_of_set {0..1::real}) |
|
4019 |
(mspace (submetric (cfunspace X euclidean_metric) |
|
78248
740b23f1138a
EXPERIMENTAL replacement of f ` A <= B by f : A -> B in Analysis
paulson <lp15@cam.ac.uk>
parents:
78202
diff
changeset
|
4020 |
{f. f \<in> topspace X \<rightarrow> {0..1}})) ) |
740b23f1138a
EXPERIMENTAL replacement of f ` A <= B by f : A -> B in Analysis
paulson <lp15@cam.ac.uk>
parents:
78202
diff
changeset
|
4021 |
(\<lambda>x. \<lambda>f \<in> mspace (submetric (cfunspace X euclidean_metric) {f. f \<in> topspace X \<rightarrow> {0..1}}). |
78202 | 4022 |
f x)" |
4023 |
proof - |
|
78248
740b23f1138a
EXPERIMENTAL replacement of f ` A <= B by f : A -> B in Analysis
paulson <lp15@cam.ac.uk>
parents:
78202
diff
changeset
|
4024 |
define K where "K \<equiv> mspace(submetric (cfunspace X euclidean_metric) {f. f \<in> topspace X \<rightarrow> {0..1::real}})" |
78202 | 4025 |
define e where "e \<equiv> \<lambda>x. \<lambda>f\<in>K. f x" |
4026 |
have "e x \<noteq> e y" if xy: "x \<noteq> y" "x \<in> topspace X" "y \<in> topspace X" for x y |
|
4027 |
proof - |
|
4028 |
have "closedin X {x}" |
|
4029 |
by (simp add: \<open>Hausdorff_space X\<close> closedin_Hausdorff_singleton \<open>x \<in> topspace X\<close>) |
|
4030 |
then obtain f where contf: "continuous_map X euclideanreal f" |
|
4031 |
and f01: "\<And>x. x \<in> topspace X \<Longrightarrow> f x \<in> {0..1}" and fxy: "f y = 0" "f x = 1" |
|
4032 |
using \<open>completely_regular_space X\<close> xy unfolding completely_regular_space_def |
|
4033 |
by (smt (verit, ccfv_threshold) Diff_iff continuous_map_in_subtopology image_subset_iff singleton_iff) |
|
4034 |
then have "bounded (f ` topspace X)" |
|
4035 |
by (meson bounded_closed_interval bounded_subset image_subset_iff) |
|
4036 |
with contf f01 have "restrict f (topspace X) \<in> K" |
|
4037 |
by (auto simp: K_def) |
|
4038 |
with fxy xy show ?thesis |
|
4039 |
unfolding e_def by (metis restrict_apply' zero_neq_one) |
|
4040 |
qed |
|
4041 |
then have "inj_on e (topspace X)" |
|
4042 |
by (meson inj_onI) |
|
4043 |
then obtain e' where e': "\<And>x. x \<in> topspace X \<Longrightarrow> e' (e x) = x" |
|
4044 |
by (metis inv_into_f_f) |
|
4045 |
have "continuous_map (subtopology (product_topology (\<lambda>f. top_of_set {0..1}) K) (e ` topspace X)) X e'" |
|
4046 |
proof (clarsimp simp add: continuous_map_atin limitin_atin openin_subtopology_alt e') |
|
4047 |
fix x U |
|
4048 |
assume "e x \<in> K \<rightarrow>\<^sub>E {0..1}" and "x \<in> topspace X" and "openin X U" and "x \<in> U" |
|
4049 |
then obtain g where contg: "continuous_map X (top_of_set {0..1}) g" and "g x = 0" |
|
4050 |
and gim: "g ` (topspace X - U) \<subseteq> {1::real}" |
|
78248
740b23f1138a
EXPERIMENTAL replacement of f ` A <= B by f : A -> B in Analysis
paulson <lp15@cam.ac.uk>
parents:
78202
diff
changeset
|
4051 |
using \<open>completely_regular_space X\<close> unfolding completely_regular_space_def |
78202 | 4052 |
by (metis Diff_iff openin_closedin_eq) |
4053 |
then have "bounded (g ` topspace X)" |
|
4054 |
by (meson bounded_closed_interval bounded_subset continuous_map_in_subtopology) |
|
78248
740b23f1138a
EXPERIMENTAL replacement of f ` A <= B by f : A -> B in Analysis
paulson <lp15@cam.ac.uk>
parents:
78202
diff
changeset
|
4055 |
moreover have "g \<in> topspace X \<rightarrow> {0..1}" |
740b23f1138a
EXPERIMENTAL replacement of f ` A <= B by f : A -> B in Analysis
paulson <lp15@cam.ac.uk>
parents:
78202
diff
changeset
|
4056 |
using contg by (simp add: continuous_map_def) |
78202 | 4057 |
ultimately have g_in_K: "restrict g (topspace X) \<in> K" |
78248
740b23f1138a
EXPERIMENTAL replacement of f ` A <= B by f : A -> B in Analysis
paulson <lp15@cam.ac.uk>
parents:
78202
diff
changeset
|
4058 |
using contg by (force simp add: K_def continuous_map_in_subtopology) |
78202 | 4059 |
have "openin (top_of_set {0..1}) {0..<1::real}" |
4060 |
using open_real_greaterThanLessThan[of "-1" 1] by (force simp: openin_open) |
|
4061 |
moreover have "e x \<in> (\<Pi>\<^sub>E f\<in>K. if f = restrict g (topspace X) then {0..<1} else {0..1})" |
|
4062 |
using \<open>e x \<in> K \<rightarrow>\<^sub>E {0..1}\<close> by (simp add: e_def \<open>g x = 0\<close> \<open>x \<in> topspace X\<close> PiE_iff) |
|
4063 |
moreover have "e y = e x" |
|
4064 |
if "y \<notin> U" and ey: "e y \<in> (\<Pi>\<^sub>E f\<in>K. if f = restrict g (topspace X) then {0..<1} else {0..1})" |
|
4065 |
and y: "y \<in> topspace X" for y |
|
4066 |
proof - |
|
4067 |
have "e y (restrict g (topspace X)) \<in> {0..<1}" |
|
4068 |
using ey by (smt (verit, ccfv_SIG) PiE_mem g_in_K) |
|
4069 |
with gim g_in_K y \<open>y \<notin> U\<close> show ?thesis |
|
4070 |
by (fastforce simp: e_def) |
|
4071 |
qed |
|
4072 |
ultimately |
|
4073 |
show "\<exists>W. openin (product_topology (\<lambda>f. top_of_set {0..1}) K) W \<and> e x \<in> W \<and> e' ` (e ` topspace X \<inter> W - {e x}) \<subseteq> U" |
|
4074 |
apply (rule_tac x="PiE K (\<lambda>f. if f = restrict g (topspace X) then {0..<1} else {0..1})" in exI) |
|
4075 |
by (auto simp: openin_PiE_gen e') |
|
4076 |
qed |
|
4077 |
with e' have "embedding_map X (product_topology (\<lambda>f. top_of_set {0..1}) K) e" |
|
4078 |
unfolding embedding_map_def homeomorphic_map_maps homeomorphic_maps_def |
|
4079 |
by (fastforce simp: e_def K_def continuous_map_in_subtopology continuous_map_componentwise) |
|
4080 |
then show ?thesis |
|
4081 |
by (simp add: K_def e_def) |
|
4082 |
qed |
|
4083 |
||
4084 |
||
4085 |
lemma completely_regular_space_cube_embedding: |
|
4086 |
fixes X :: "'a topology" |
|
4087 |
assumes "completely_regular_space X" "Hausdorff_space X" |
|
4088 |
obtains K:: "('a\<Rightarrow>real)set" and e |
|
4089 |
where "embedding_map X (product_topology (\<lambda>f. top_of_set {0..1::real}) K) e" |
|
4090 |
using completely_regular_space_cube_embedding_explicit [OF assms] by metis |
|
4091 |
||
4092 |
||
4093 |
subsection \<open>Urysohn and Tietze analogs for completely regular spaces\<close> |
|
4094 |
||
4095 |
text\<open>"Urysohn and Tietze analogs for completely regular spaces if (()) set is |
|
4096 |
assumed compact instead of closed. Note that Hausdorffness is *not* required: inside () proof |
|
4097 |
we factor through the Kolmogorov quotient." -- John Harrison\<close> |
|
4098 |
||
4099 |
lemma Urysohn_completely_regular_closed_compact: |
|
4100 |
fixes a b::real |
|
4101 |
assumes "a \<le> b" "completely_regular_space X" "closedin X S" "compactin X T" "disjnt S T" |
|
4102 |
obtains f where "continuous_map X (subtopology euclidean {a..b}) f" "f ` T \<subseteq> {a}" "f ` S \<subseteq> {b}" |
|
4103 |
proof - |
|
4104 |
obtain f where contf: "continuous_map X (subtopology euclideanreal {0..1}) f" |
|
4105 |
and f0: "f ` T \<subseteq> {0}" and f1: "f ` S \<subseteq> {1}" |
|
4106 |
proof (cases "T={}") |
|
4107 |
case True |
|
4108 |
show thesis |
|
4109 |
proof |
|
4110 |
show "continuous_map X (top_of_set {0..1}) (\<lambda>x. 1::real)" "(\<lambda>x. 1::real) ` T \<subseteq> {0}" "(\<lambda>x. 1::real) ` S \<subseteq> {1}" |
|
4111 |
using True by auto |
|
4112 |
qed |
|
4113 |
next |
|
4114 |
case False |
|
4115 |
have "\<And>t. t \<in> T \<Longrightarrow> \<exists>f. continuous_map X (subtopology euclideanreal ({0..1})) f \<and> f t = 0 \<and> f ` S \<subseteq> {1}" |
|
4116 |
using assms unfolding completely_regular_space_def |
|
4117 |
by (meson DiffI compactin_subset_topspace disjnt_iff subset_eq) |
|
4118 |
then obtain g where contg: "\<And>t. t \<in> T \<Longrightarrow> continuous_map X (subtopology euclideanreal {0..1}) (g t)" |
|
4119 |
and g0: "\<And>t. t \<in> T \<Longrightarrow> g t t = 0" |
|
4120 |
and g1: "\<And>t. t \<in> T \<Longrightarrow> g t ` S \<subseteq> {1}" |
|
4121 |
by metis |
|
4122 |
then have g01: "\<And>t. t \<in> T \<Longrightarrow> g t ` topspace X \<subseteq> {0..1}" |
|
4123 |
by (meson continuous_map_in_subtopology) |
|
4124 |
define G where "G \<equiv> \<lambda>t. {x \<in> topspace X. g t x \<in> {..<1/2}}" |
|
4125 |
have "Ball (G`T) (openin X)" |
|
4126 |
using contg unfolding G_def continuous_map_in_subtopology |
|
4127 |
by (smt (verit, best) Collect_cong openin_continuous_map_preimage image_iff open_lessThan open_openin) |
|
4128 |
moreover have "T \<subseteq> \<Union>(G`T)" |
|
4129 |
using \<open>compactin X T\<close> g0 compactin_subset_topspace by (force simp: G_def) |
|
4130 |
ultimately have "\<exists>\<F>. finite \<F> \<and> \<F> \<subseteq> G`T \<and> T \<subseteq> \<Union> \<F>" |
|
4131 |
using \<open>compactin X T\<close> unfolding compactin_def by blast |
|
4132 |
then obtain K where K: "finite K" "K \<subseteq> T" "T \<subseteq> \<Union>(G`K)" |
|
4133 |
by (metis finite_subset_image) |
|
4134 |
with False have "K \<noteq> {}" |
|
4135 |
by fastforce |
|
4136 |
define f where "f \<equiv> \<lambda>x. 2 * max 0 (Inf ((\<lambda>t. g t x) ` K) - 1/2)" |
|
4137 |
have [simp]: "max 0 (x - 1/2) = 0 \<longleftrightarrow> x \<le> 1/2" for x::real |
|
4138 |
by force |
|
4139 |
have [simp]: "2 * max 0 (x - 1/2) = 1 \<longleftrightarrow> x = 1" for x::real |
|
4140 |
by (simp add: max_def_raw) |
|
4141 |
show thesis |
|
4142 |
proof |
|
4143 |
have "g t s = 1" if "s \<in> S" "t \<in> K" for s t |
|
4144 |
using \<open>K \<subseteq> T\<close> g1 that by auto |
|
4145 |
then show "f ` S \<subseteq> {1}" |
|
4146 |
using \<open>K \<noteq> {}\<close> by (simp add: f_def image_subset_iff) |
|
4147 |
have "(INF t\<in>K. g t x) \<le> 1/2" if "x \<in> T" for x |
|
4148 |
proof - |
|
4149 |
obtain k where "k \<in> K" "g k x < 1/2" |
|
4150 |
using K \<open>x \<in> T\<close> by (auto simp: G_def) |
|
4151 |
then show ?thesis |
|
4152 |
by (meson \<open>finite K\<close> cInf_le_finite dual_order.trans finite_imageI imageI less_le_not_le) |
|
4153 |
qed |
|
4154 |
then show "f ` T \<subseteq> {0}" |
|
4155 |
by (force simp: f_def) |
|
4156 |
have "\<And>t. t \<in> K \<Longrightarrow> continuous_map X euclideanreal (g t)" |
|
4157 |
using \<open>K \<subseteq> T\<close> contg continuous_map_in_subtopology by blast |
|
4158 |
moreover have "2 * max 0 ((INF t\<in>K. g t x) - 1/2) \<le> 1" if "x \<in> topspace X" for x |
|
4159 |
proof - |
|
4160 |
obtain k where "k \<in> K" "g k x \<le> 1" |
|
4161 |
using K \<open>x \<in> topspace X\<close> \<open>K \<noteq> {}\<close> g01 by (fastforce simp: G_def) |
|
4162 |
then have "(INF t\<in>K. g t x) \<le> 1" |
|
4163 |
by (meson \<open>finite K\<close> cInf_le_finite dual_order.trans finite_imageI imageI) |
|
4164 |
then show ?thesis |
|
4165 |
by (simp add: max_def_raw) |
|
4166 |
qed |
|
4167 |
ultimately show "continuous_map X (top_of_set {0..1}) f" |
|
4168 |
by (force simp: f_def continuous_map_in_subtopology intro!: \<open>finite K\<close> continuous_intros) |
|
4169 |
qed |
|
4170 |
qed |
|
4171 |
define g where "g \<equiv> \<lambda>x. a + (b - a) * f x" |
|
4172 |
show thesis |
|
4173 |
proof |
|
4174 |
have "\<forall>x\<in>topspace X. a + (b - a) * f x \<le> b" |
|
4175 |
using contf \<open>a \<le> b\<close> apply (simp add: continuous_map_in_subtopology image_subset_iff) |
|
4176 |
by (smt (verit, best) mult_right_le_one_le) |
|
4177 |
then show "continuous_map X (top_of_set {a..b}) g" |
|
4178 |
using contf \<open>a \<le> b\<close> unfolding g_def continuous_map_in_subtopology image_subset_iff |
|
4179 |
by (intro conjI continuous_intros; simp) |
|
4180 |
show "g ` T \<subseteq> {a}" "g ` S \<subseteq> {b}" |
|
4181 |
using f0 f1 by (auto simp: g_def) |
|
4182 |
qed |
|
4183 |
qed |
|
4184 |
||
4185 |
||
4186 |
lemma Urysohn_completely_regular_compact_closed: |
|
4187 |
fixes a b::real |
|
4188 |
assumes "a \<le> b" "completely_regular_space X" "compactin X S" "closedin X T" "disjnt S T" |
|
4189 |
obtains f where "continuous_map X (subtopology euclidean {a..b}) f" "f ` T \<subseteq> {a}" "f ` S \<subseteq> {b}" |
|
4190 |
proof - |
|
4191 |
obtain f where contf: "continuous_map X (subtopology euclidean {-b..-a}) f" and fim: "f ` T \<subseteq> {-a}" "f ` S \<subseteq> {-b}" |
|
4192 |
by (meson Urysohn_completely_regular_closed_compact assms disjnt_sym neg_le_iff_le) |
|
4193 |
show thesis |
|
4194 |
proof |
|
4195 |
show "continuous_map X (top_of_set {a..b}) (uminus \<circ> f)" |
|
4196 |
using contf by (auto simp: continuous_map_in_subtopology o_def) |
|
4197 |
show "(uminus o f) ` T \<subseteq> {a}" "(uminus o f) ` S \<subseteq> {b}" |
|
4198 |
using fim by fastforce+ |
|
4199 |
qed |
|
4200 |
qed |
|
4201 |
||
4202 |
lemma Urysohn_completely_regular_compact_closed_alt: |
|
4203 |
fixes a b::real |
|
4204 |
assumes "completely_regular_space X" "compactin X S" "closedin X T" "disjnt S T" |
|
4205 |
obtains f where "continuous_map X euclideanreal f" "f ` T \<subseteq> {a}" "f ` S \<subseteq> {b}" |
|
4206 |
proof (cases a b rule: le_cases) |
|
4207 |
case le |
|
4208 |
then show ?thesis |
|
4209 |
by (meson Urysohn_completely_regular_compact_closed assms continuous_map_into_fulltopology that) |
|
4210 |
next |
|
4211 |
case ge |
|
4212 |
then show ?thesis |
|
4213 |
using Urysohn_completely_regular_closed_compact assms |
|
4214 |
by (metis Urysohn_completely_regular_closed_compact assms continuous_map_into_fulltopology disjnt_sym that) |
|
4215 |
qed |
|
4216 |
||
4217 |
||
4218 |
lemma Tietze_extension_comp_reg_aux: |
|
4219 |
fixes T :: "real set" |
|
4220 |
assumes "completely_regular_space X" "Hausdorff_space X" "compactin X S" |
|
4221 |
and T: "is_interval T" "T\<noteq>{}" |
|
4222 |
and contf: "continuous_map (subtopology X S) euclidean f" and fim: "f`S \<subseteq> T" |
|
4223 |
obtains g where "continuous_map X euclidean g" "g ` topspace X \<subseteq> T" "\<And>x. x \<in> S \<Longrightarrow> g x = f x" |
|
4224 |
proof - |
|
4225 |
obtain K:: "('a\<Rightarrow>real)set" and e |
|
4226 |
where e0: "embedding_map X (product_topology (\<lambda>f. top_of_set {0..1::real}) K) e" |
|
4227 |
using assms completely_regular_space_cube_embedding by blast |
|
4228 |
define cube where "cube \<equiv> product_topology (\<lambda>f. top_of_set {0..1::real}) K" |
|
4229 |
have e: "embedding_map X cube e" |
|
4230 |
using e0 by (simp add: cube_def) |
|
4231 |
obtain e' where e': "homeomorphic_maps X (subtopology cube (e ` topspace X)) e e'" |
|
4232 |
using e by (force simp: cube_def embedding_map_def homeomorphic_map_maps) |
|
4233 |
then have conte: "continuous_map X (subtopology cube (e ` topspace X)) e" |
|
4234 |
and conte': "continuous_map (subtopology cube (e ` topspace X)) X e'" |
|
4235 |
and e'e: "\<forall>x\<in>topspace X. e' (e x) = x" |
|
4236 |
by (auto simp: homeomorphic_maps_def) |
|
4237 |
have "Hausdorff_space cube" |
|
4238 |
unfolding cube_def |
|
4239 |
using Hausdorff_space_euclidean Hausdorff_space_product_topology Hausdorff_space_subtopology by blast |
|
4240 |
have "normal_space cube" |
|
4241 |
proof (rule compact_Hausdorff_or_regular_imp_normal_space) |
|
4242 |
show "compact_space cube" |
|
4243 |
unfolding cube_def |
|
4244 |
using compact_space_product_topology compact_space_subtopology compactin_euclidean_iff by blast |
|
4245 |
qed (use \<open>Hausdorff_space cube\<close> in auto) |
|
4246 |
moreover |
|
4247 |
have comp: "compactin cube (e ` S)" |
|
4248 |
by (meson \<open>compactin X S\<close> conte continuous_map_in_subtopology image_compactin) |
|
4249 |
then have "closedin cube (e ` S)" |
|
4250 |
by (intro compactin_imp_closedin \<open>Hausdorff_space cube\<close>) |
|
4251 |
moreover |
|
4252 |
have "continuous_map (subtopology cube (e ` S)) euclideanreal (f \<circ> e')" |
|
4253 |
proof (intro continuous_map_compose) |
|
4254 |
show "continuous_map (subtopology cube (e ` S)) (subtopology X S) e'" |
|
4255 |
unfolding continuous_map_in_subtopology |
|
4256 |
proof |
|
4257 |
show "continuous_map (subtopology cube (e ` S)) X e'" |
|
4258 |
by (meson \<open>compactin X S\<close> compactin_subset_topspace conte' continuous_map_from_subtopology_mono image_mono) |
|
4259 |
show "e' ` topspace (subtopology cube (e ` S)) \<subseteq> S" |
|
4260 |
using \<open>compactin X S\<close> compactin_subset_topspace e'e by fastforce |
|
4261 |
qed |
|
4262 |
qed (simp add: contf) |
|
4263 |
moreover |
|
4264 |
have "(f \<circ> e') ` e ` S \<subseteq> T" |
|
4265 |
using \<open>compactin X S\<close> compactin_subset_topspace e'e fim by fastforce |
|
4266 |
ultimately |
|
4267 |
obtain g where contg: "continuous_map cube euclidean g" and gsub: "g ` topspace cube \<subseteq> T" |
|
4268 |
and gf: "\<And>x. x \<in> e`S \<Longrightarrow> g x = (f \<circ> e') x" |
|
4269 |
using Tietze_extension_realinterval T by metis |
|
4270 |
show thesis |
|
4271 |
proof |
|
4272 |
show "continuous_map X euclideanreal (g \<circ> e)" |
|
4273 |
by (meson contg conte continuous_map_compose continuous_map_in_subtopology) |
|
4274 |
show "(g \<circ> e) ` topspace X \<subseteq> T" |
|
4275 |
using gsub conte continuous_map_image_subset_topspace by fastforce |
|
4276 |
fix x |
|
4277 |
assume "x \<in> S" |
|
4278 |
then show "(g \<circ> e) x = f x" |
|
4279 |
using gf \<open>compactin X S\<close> compactin_subset_topspace e'e by fastforce |
|
4280 |
qed |
|
4281 |
qed |
|
4282 |
||
4283 |
||
4284 |
lemma Tietze_extension_completely_regular: |
|
4285 |
assumes "completely_regular_space X" "compactin X S" "is_interval T" "T \<noteq> {}" |
|
4286 |
and contf: "continuous_map (subtopology X S) euclidean f" and fim: "f`S \<subseteq> T" |
|
4287 |
obtains g where "continuous_map X euclideanreal g" "g ` topspace X \<subseteq> T" |
|
4288 |
"\<And>x. x \<in> S \<Longrightarrow> g x = f x" |
|
4289 |
proof - |
|
4290 |
define Q where "Q \<equiv> Kolmogorov_quotient X ` (topspace X)" |
|
4291 |
obtain g where contg: "continuous_map (subtopology X (Kolmogorov_quotient X ` S)) euclidean g" |
|
4292 |
and gf: "\<And>x. x \<in> S \<Longrightarrow> g(Kolmogorov_quotient X x) = f x" |
|
4293 |
using Kolmogorov_quotient_lift_exists |
|
4294 |
by (metis \<open>compactin X S\<close> contf compactin_subset_topspace open_openin t0_space_def t1_space) |
|
4295 |
have "S \<subseteq> topspace X" |
|
4296 |
by (simp add: \<open>compactin X S\<close> compactin_subset_topspace) |
|
4297 |
then have [simp]: "Q \<inter> Kolmogorov_quotient X ` S = Kolmogorov_quotient X ` S" |
|
4298 |
using Q_def by blast |
|
4299 |
have creg: "completely_regular_space (subtopology X Q)" |
|
4300 |
by (simp add: \<open>completely_regular_space X\<close> completely_regular_space_subtopology) |
|
4301 |
then have "regular_space (subtopology X Q)" |
|
4302 |
by (simp add: completely_regular_imp_regular_space) |
|
4303 |
then have "Hausdorff_space (subtopology X Q)" |
|
4304 |
using Q_def regular_t0_eq_Hausdorff_space t0_space_Kolmogorov_quotient by blast |
|
4305 |
moreover |
|
4306 |
have "compactin (subtopology X Q) (Kolmogorov_quotient X ` S)" |
|
4307 |
by (metis Q_def \<open>compactin X S\<close> image_compactin quotient_imp_continuous_map quotient_map_Kolmogorov_quotient) |
|
4308 |
ultimately obtain h where conth: "continuous_map (subtopology X Q) euclidean h" |
|
4309 |
and him: "h ` topspace (subtopology X Q) \<subseteq> T" |
|
4310 |
and hg: "\<And>x. x \<in> Kolmogorov_quotient X ` S \<Longrightarrow> h x = g x" |
|
4311 |
using Tietze_extension_comp_reg_aux [of "subtopology X Q" "Kolmogorov_quotient X ` S" T g] |
|
4312 |
apply (simp add: subtopology_subtopology creg contg assms) |
|
4313 |
using fim gf by blast |
|
4314 |
show thesis |
|
4315 |
proof |
|
4316 |
show "continuous_map X euclideanreal (h \<circ> Kolmogorov_quotient X)" |
|
4317 |
by (metis Q_def conth continuous_map_compose quotient_imp_continuous_map quotient_map_Kolmogorov_quotient) |
|
4318 |
show "(h \<circ> Kolmogorov_quotient X) ` topspace X \<subseteq> T" |
|
4319 |
using Q_def continuous_map_Kolmogorov_quotient continuous_map_image_subset_topspace him by fastforce |
|
4320 |
fix x |
|
4321 |
assume "x \<in> S" then show "(h \<circ> Kolmogorov_quotient X) x = f x" |
|
4322 |
by (simp add: gf hg) |
|
4323 |
qed |
|
4324 |
qed |
|
4325 |
||
78250
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4326 |
subsection\<open>Size bounds on connected or path-connected spaces\<close> |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4327 |
|
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4328 |
lemma connected_space_imp_card_ge_alt: |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4329 |
assumes "connected_space X" "completely_regular_space X" "closedin X S" "S \<noteq> {}" "S \<noteq> topspace X" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4330 |
shows "(UNIV::real set) \<lesssim> topspace X" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4331 |
proof - |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4332 |
have "S \<subseteq> topspace X" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4333 |
using \<open>closedin X S\<close> closedin_subset by blast |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4334 |
then obtain a where "a \<in> topspace X" "a \<notin> S" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4335 |
using \<open>S \<noteq> topspace X\<close> by blast |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4336 |
have "(UNIV::real set) \<lesssim> {0..1::real}" |
78256
71e1aa0d9421
A couple of new lemmas involving cardinality
paulson <lp15@cam.ac.uk>
parents:
78250
diff
changeset
|
4337 |
using eqpoll_real_subset |
78250
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4338 |
by (meson atLeastAtMost_iff eqpoll_imp_lepoll eqpoll_sym less_eq_real_def zero_less_one) |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4339 |
also have "\<dots> \<lesssim> topspace X" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4340 |
proof - |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4341 |
obtain f where contf: "continuous_map X euclidean f" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4342 |
and fim: "f \<in> (topspace X) \<rightarrow> {0..1::real}" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4343 |
and f0: "f a = 0" and f1: "f ` S \<subseteq> {1}" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4344 |
using \<open>completely_regular_space X\<close> |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4345 |
unfolding completely_regular_space_def |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4346 |
by (metis Diff_iff \<open>a \<in> topspace X\<close> \<open>a \<notin> S\<close> \<open>closedin X S\<close> continuous_map_in_subtopology image_subset_iff_funcset) |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4347 |
have "\<exists>y\<in>topspace X. x = f y" if "0 \<le> x" and "x \<le> 1" for x |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4348 |
proof - |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4349 |
have "connectedin euclidean (f ` topspace X)" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4350 |
using \<open>connected_space X\<close> connectedin_continuous_map_image connectedin_topspace contf by blast |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4351 |
moreover have "\<exists>y. 0 = f y \<and> y \<in> topspace X" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4352 |
using \<open>a \<in> topspace X\<close> f0 by auto |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4353 |
moreover have "\<exists>y. 1 = f y \<and> y \<in> topspace X" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4354 |
using \<open>S \<subseteq> topspace X\<close> \<open>S \<noteq> {}\<close> f1 by fastforce |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4355 |
ultimately show ?thesis |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4356 |
using that by (fastforce simp: is_interval_1 simp flip: is_interval_connected_1) |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4357 |
qed |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4358 |
then show ?thesis |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4359 |
unfolding lepoll_iff using atLeastAtMost_iff by blast |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4360 |
qed |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4361 |
finally show ?thesis . |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4362 |
qed |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4363 |
|
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4364 |
|
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4365 |
lemma connected_space_imp_card_ge_gen: |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4366 |
assumes "connected_space X" "normal_space X" "closedin X S" "closedin X T" "S \<noteq> {}" "T \<noteq> {}" "disjnt S T" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4367 |
shows "(UNIV::real set) \<lesssim> topspace X" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4368 |
proof - |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4369 |
have "(UNIV::real set) \<lesssim> {0..1::real}" |
78256
71e1aa0d9421
A couple of new lemmas involving cardinality
paulson <lp15@cam.ac.uk>
parents:
78250
diff
changeset
|
4370 |
by (metis atLeastAtMost_iff eqpoll_real_subset eqpoll_imp_lepoll eqpoll_sym less_le_not_le zero_less_one) |
78250
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4371 |
also have "\<dots>\<lesssim> topspace X" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4372 |
proof - |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4373 |
obtain f where contf: "continuous_map X euclidean f" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4374 |
and fim: "f \<in> (topspace X) \<rightarrow> {0..1::real}" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4375 |
and f0: "f ` S \<subseteq> {0}" and f1: "f ` T \<subseteq> {1}" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4376 |
using assms by (metis continuous_map_in_subtopology normal_space_iff_Urysohn image_subset_iff_funcset) |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4377 |
have "\<exists>y\<in>topspace X. x = f y" if "0 \<le> x" and "x \<le> 1" for x |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4378 |
proof - |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4379 |
have "connectedin euclidean (f ` topspace X)" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4380 |
using \<open>connected_space X\<close> connectedin_continuous_map_image connectedin_topspace contf by blast |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4381 |
moreover have "\<exists>y. 0 = f y \<and> y \<in> topspace X" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4382 |
using \<open>closedin X S\<close> \<open>S \<noteq> {}\<close> closedin_subset f0 by fastforce |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4383 |
moreover have "\<exists>y. 1 = f y \<and> y \<in> topspace X" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4384 |
using \<open>closedin X T\<close> \<open>T \<noteq> {}\<close> closedin_subset f1 by fastforce |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4385 |
ultimately show ?thesis |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4386 |
using that by (fastforce simp: is_interval_1 simp flip: is_interval_connected_1) |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4387 |
qed |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4388 |
then show ?thesis |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4389 |
unfolding lepoll_iff using atLeastAtMost_iff by blast |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4390 |
qed |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4391 |
finally show ?thesis . |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4392 |
qed |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4393 |
|
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4394 |
lemma connected_space_imp_card_ge: |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4395 |
assumes "connected_space X" "normal_space X" "t1_space X" and nosing: "\<not> (\<exists>a. topspace X \<subseteq> {a})" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4396 |
shows "(UNIV::real set) \<lesssim> topspace X" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4397 |
proof - |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4398 |
obtain a b where "a \<in> topspace X" "b \<in> topspace X" "a \<noteq> b" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4399 |
by (metis nosing singletonI subset_iff) |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4400 |
then have "{a} \<noteq> topspace X" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4401 |
by force |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4402 |
with connected_space_imp_card_ge_alt assms show ?thesis |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4403 |
by (metis \<open>a \<in> topspace X\<close> closedin_t1_singleton insert_not_empty normal_imp_completely_regular_space_A) |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4404 |
qed |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4405 |
|
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4406 |
lemma connected_space_imp_infinite_gen: |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4407 |
"\<lbrakk>connected_space X; t1_space X; \<nexists>a. topspace X \<subseteq> {a}\<rbrakk> \<Longrightarrow> infinite(topspace X)" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4408 |
by (metis connected_space_discrete_topology finite_t1_space_imp_discrete_topology) |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4409 |
|
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4410 |
lemma connected_space_imp_infinite: |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4411 |
"\<lbrakk>connected_space X; Hausdorff_space X; \<nexists>a. topspace X \<subseteq> {a}\<rbrakk> \<Longrightarrow> infinite(topspace X)" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4412 |
by (simp add: Hausdorff_imp_t1_space connected_space_imp_infinite_gen) |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4413 |
|
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4414 |
lemma connected_space_imp_infinite_alt: |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4415 |
assumes "connected_space X" "regular_space X" "closedin X S" "S \<noteq> {}" "S \<noteq> topspace X" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4416 |
shows "infinite(topspace X)" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4417 |
proof - |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4418 |
have "S \<subseteq> topspace X" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4419 |
using \<open>closedin X S\<close> closedin_subset by blast |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4420 |
then obtain a where a: "a \<in> topspace X" "a \<notin> S" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4421 |
using \<open>S \<noteq> topspace X\<close> by blast |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4422 |
have "\<exists>\<Phi>. \<forall>n. (disjnt (\<Phi> n) S \<and> a \<in> \<Phi> n \<and> openin X (\<Phi> n)) \<and> \<Phi>(Suc n) \<subset> \<Phi> n" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4423 |
proof (rule dependent_nat_choice) |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4424 |
show "\<exists>T. disjnt T S \<and> a \<in> T \<and> openin X T" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4425 |
by (metis Diff_iff a \<open>closedin X S\<close> closedin_def disjnt_iff) |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4426 |
fix V n |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4427 |
assume \<section>: "disjnt V S \<and> a \<in> V \<and> openin X V" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4428 |
then obtain U C where U: "openin X U" "closedin X C" "a \<in> U" "U \<subseteq> C" "C \<subseteq> V" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4429 |
using \<open>regular_space X\<close> by (metis neighbourhood_base_of neighbourhood_base_of_closedin) |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4430 |
with assms have "U \<subset> V" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4431 |
by (metis "\<section>" \<open>S \<subseteq> topspace X\<close> connected_space_clopen_in disjnt_def empty_iff inf.absorb_iff2 inf.orderE psubsetI subset_trans) |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4432 |
with U show "\<exists>U. (disjnt U S \<and> a \<in> U \<and> openin X U) \<and> U \<subset> V" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4433 |
using "\<section>" disjnt_subset1 by blast |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4434 |
qed |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4435 |
then obtain \<Phi> where \<Phi>: "\<And>n. disjnt (\<Phi> n) S \<and> a \<in> \<Phi> n \<and> openin X (\<Phi> n)" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4436 |
and \<Phi>sub: "\<And>n. \<Phi> (Suc n) \<subset> \<Phi> n" by metis |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4437 |
then have "decseq \<Phi>" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4438 |
by (simp add: decseq_SucI psubset_eq) |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4439 |
have "\<forall>n. \<exists>x. x \<in> \<Phi> n \<and> x \<notin> \<Phi>(Suc n)" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4440 |
by (meson \<Phi>sub psubsetE subsetI) |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4441 |
then obtain f where fin: "\<And>n. f n \<in> \<Phi> n" and fout: "\<And>n. f n \<notin> \<Phi>(Suc n)" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4442 |
by metis |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4443 |
have "range f \<subseteq> topspace X" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4444 |
by (meson \<Phi> fin image_subset_iff openin_subset subset_iff) |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4445 |
moreover have "inj f" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4446 |
by (metis Suc_le_eq \<open>decseq \<Phi>\<close> decseq_def fin fout linorder_injI subsetD) |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4447 |
ultimately show ?thesis |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4448 |
using infinite_iff_countable_subset by blast |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4449 |
qed |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4450 |
|
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4451 |
lemma path_connected_space_imp_card_ge: |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4452 |
assumes "path_connected_space X" "Hausdorff_space X" and nosing: "\<not> (\<exists>x. topspace X \<subseteq> {x})" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4453 |
shows "(UNIV::real set) \<lesssim> topspace X" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4454 |
proof - |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4455 |
obtain a b where "a \<in> topspace X" "b \<in> topspace X" "a \<noteq> b" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4456 |
by (metis nosing singletonI subset_iff) |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4457 |
then obtain \<gamma> where \<gamma>: "pathin X \<gamma>" "\<gamma> 0 = a" "\<gamma> 1 = b" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4458 |
by (meson \<open>a \<in> topspace X\<close> \<open>b \<in> topspace X\<close> \<open>path_connected_space X\<close> path_connected_space_def) |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4459 |
let ?Y = "subtopology X (\<gamma> ` (topspace (subtopology euclidean {0..1})))" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4460 |
have "(UNIV::real set) \<lesssim> topspace ?Y" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4461 |
proof (intro compact_Hausdorff_or_regular_imp_normal_space connected_space_imp_card_ge) |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4462 |
show "connected_space ?Y" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4463 |
using \<open>pathin X \<gamma>\<close> connectedin_def connectedin_path_image by auto |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4464 |
show "Hausdorff_space ?Y \<or> regular_space ?Y" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4465 |
using Hausdorff_space_subtopology \<open>Hausdorff_space X\<close> by blast |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4466 |
show "t1_space ?Y" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4467 |
using Hausdorff_imp_t1_space \<open>Hausdorff_space X\<close> t1_space_subtopology by blast |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4468 |
show "compact_space ?Y" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4469 |
by (simp add: \<open>pathin X \<gamma>\<close> compact_space_subtopology compactin_path_image) |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4470 |
have "a \<in> topspace ?Y" "b \<in> topspace ?Y" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4471 |
using \<gamma> pathin_subtopology by fastforce+ |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4472 |
with \<open>a \<noteq> b\<close> show "\<nexists>x. topspace ?Y \<subseteq> {x}" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4473 |
by blast |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4474 |
qed |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4475 |
also have "\<dots> \<lesssim> \<gamma> ` {0..1}" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4476 |
by (simp add: subset_imp_lepoll) |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4477 |
also have "\<dots> \<lesssim> topspace X" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4478 |
by (meson \<gamma> path_image_subset_topspace subset_imp_lepoll image_subset_iff_funcset) |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4479 |
finally show ?thesis . |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4480 |
qed |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4481 |
|
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4482 |
lemma connected_space_imp_uncountable: |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4483 |
assumes "connected_space X" "regular_space X" "Hausdorff_space X" "\<not> (\<exists>a. topspace X \<subseteq> {a})" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4484 |
shows "\<not> countable(topspace X)" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4485 |
proof |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4486 |
assume coX: "countable (topspace X)" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4487 |
with \<open>regular_space X\<close> have "normal_space X" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4488 |
using countable_imp_Lindelof_space regular_Lindelof_imp_normal_space by blast |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4489 |
then have "(UNIV::real set) \<lesssim> topspace X" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4490 |
by (simp add: Hausdorff_imp_t1_space assms connected_space_imp_card_ge) |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4491 |
with coX show False |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4492 |
using countable_lepoll uncountable_UNIV_real by blast |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4493 |
qed |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4494 |
|
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4495 |
lemma path_connected_space_imp_uncountable: |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4496 |
assumes "path_connected_space X" "t1_space X" and nosing: "\<not> (\<exists>a. topspace X \<subseteq> {a})" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4497 |
shows "\<not> countable(topspace X)" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4498 |
proof |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4499 |
assume coX: "countable (topspace X)" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4500 |
obtain a b where "a \<in> topspace X" "b \<in> topspace X" "a \<noteq> b" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4501 |
by (metis nosing singletonI subset_iff) |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4502 |
then obtain \<gamma> where "pathin X \<gamma>" "\<gamma> 0 = a" "\<gamma> 1 = b" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4503 |
by (meson \<open>a \<in> topspace X\<close> \<open>b \<in> topspace X\<close> \<open>path_connected_space X\<close> path_connected_space_def) |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4504 |
then have "\<gamma> ` {0..1} \<lesssim> topspace X" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4505 |
by (meson path_image_subset_topspace subset_imp_lepoll image_subset_iff_funcset) |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4506 |
define \<A> where "\<A> \<equiv> ((\<lambda>a. {x \<in> {0..1}. \<gamma> x \<in> {a}}) ` topspace X) - {{}}" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4507 |
have \<A>01: "\<A> = {{0..1}}" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4508 |
proof (rule real_Sierpinski_lemma) |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4509 |
show "countable \<A>" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4510 |
using \<A>_def coX by blast |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4511 |
show "disjoint \<A>" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4512 |
by (auto simp: \<A>_def disjnt_iff pairwise_def) |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4513 |
show "\<Union>\<A> = {0..1}" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4514 |
using \<open>pathin X \<gamma>\<close> path_image_subset_topspace by (fastforce simp: \<A>_def Bex_def) |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4515 |
fix C |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4516 |
assume "C \<in> \<A>" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4517 |
then obtain a where "a \<in> topspace X" and C: "C = {x \<in> {0..1}. \<gamma> x \<in> {a}}" "C \<noteq> {}" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4518 |
by (auto simp: \<A>_def) |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4519 |
then have "closedin X {a}" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4520 |
by (meson \<open>t1_space X\<close> closedin_t1_singleton) |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4521 |
then have "closedin (top_of_set {0..1}) C" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4522 |
using C \<open>pathin X \<gamma>\<close> closedin_continuous_map_preimage pathin_def by fastforce |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4523 |
then show "closed C \<and> C \<noteq> {}" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4524 |
using C closedin_closed_trans by blast |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4525 |
qed auto |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4526 |
then have "{0..1} \<in> \<A>" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4527 |
by blast |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4528 |
then have "\<exists>a \<in> topspace X. {0..1} \<subseteq> {x. \<gamma> x = a}" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4529 |
using \<A>_def image_iff by auto |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4530 |
then show False |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4531 |
using \<open>\<gamma> 0 = a\<close> \<open>\<gamma> 1 = b\<close> \<open>a \<noteq> b\<close> atLeastAtMost_iff zero_less_one_class.zero_le_one by blast |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4532 |
qed |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4533 |
|
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4534 |
|
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4535 |
subsection\<open>Lavrentiev extension etc\<close> |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4536 |
|
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4537 |
lemma (in Metric_space) convergent_eq_zero_oscillation_gen: |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4538 |
assumes "mcomplete" and fim: "f \<in> (topspace X \<inter> S) \<rightarrow> M" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4539 |
shows "(\<exists>l. limitin mtopology f l (atin_within X a S)) \<longleftrightarrow> |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4540 |
M \<noteq> {} \<and> |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4541 |
(a \<in> topspace X |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4542 |
\<longrightarrow> (\<forall>\<epsilon>>0. \<exists>U. openin X U \<and> a \<in> U \<and> |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4543 |
(\<forall>x \<in> (S \<inter> U) - {a}. \<forall>y \<in> (S \<inter> U) - {a}. d (f x) (f y) < \<epsilon>)))" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4544 |
proof (cases "M = {}") |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4545 |
case True |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4546 |
with limitin_mspace show ?thesis |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4547 |
by blast |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4548 |
next |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4549 |
case False |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4550 |
show ?thesis |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4551 |
proof (cases "a \<in> topspace X") |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4552 |
case True |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4553 |
let ?R = "\<forall>\<epsilon>>0. \<exists>U. openin X U \<and> a \<in> U \<and> (\<forall>x\<in>S \<inter> U - {a}. \<forall>y\<in>S \<inter> U - {a}. d (f x) (f y) < \<epsilon>)" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4554 |
show ?thesis |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4555 |
proof (cases "a \<in> X derived_set_of S") |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4556 |
case True |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4557 |
have ?R |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4558 |
if "limitin mtopology f l (atin_within X a S)" for l |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4559 |
proof (intro strip) |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4560 |
fix \<epsilon>::real |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4561 |
assume "\<epsilon>>0" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4562 |
with that \<open>a \<in> topspace X\<close> |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4563 |
obtain U where U: "openin X U" "a \<in> U" "l \<in> M" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4564 |
and Uless: "\<forall>x\<in>U - {a}. x \<in> S \<longrightarrow> f x \<in> M \<and> d (f x) l < \<epsilon>/2" |
78750
f229090cb8a3
atin now an abbreviation for atin_within, which has been moved to Abstract_Limits
paulson <lp15@cam.ac.uk>
parents:
78336
diff
changeset
|
4565 |
unfolding limitin_metric eventually_atin_within by (metis Diff_iff insertI1 half_gt_zero [OF \<open>\<epsilon>>0\<close>]) |
78250
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4566 |
show "\<exists>U. openin X U \<and> a \<in> U \<and> (\<forall>x\<in>S \<inter> U - {a}. \<forall>y\<in>S \<inter> U - {a}. d (f x) (f y) < \<epsilon>)" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4567 |
proof (intro exI strip conjI) |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4568 |
fix x y |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4569 |
assume x: "x \<in> S \<inter> U - {a}" and y: "y \<in> S \<inter> U - {a}" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4570 |
then have "d (f x) l < \<epsilon>/2" "d (f y) l < \<epsilon>/2" "f x \<in> M" "f y \<in> M" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4571 |
using Uless by auto |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4572 |
then show "d (f x) (f y) < \<epsilon>" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4573 |
using triangle' \<open>l \<in> M\<close> by fastforce |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4574 |
qed (auto simp add: U) |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4575 |
qed |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4576 |
moreover have "\<exists>l. limitin mtopology f l (atin_within X a S)" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4577 |
if R [rule_format]: ?R |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4578 |
proof - |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4579 |
define F where "F \<equiv> \<lambda>U. mtopology closure_of f ` (S \<inter> U - {a})" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4580 |
define \<C> where "\<C> \<equiv> F ` {U. openin X U \<and> a \<in> U}" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4581 |
have \<C>_clo: "\<forall>C \<in> \<C>. closedin mtopology C" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4582 |
by (force simp add: \<C>_def F_def) |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4583 |
moreover have sub_mcball: "\<exists>C a. C \<in> \<C> \<and> C \<subseteq> mcball a \<epsilon>" if "\<epsilon>>0" for \<epsilon> |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4584 |
proof - |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4585 |
obtain U where U: "openin X U" "a \<in> U" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4586 |
and Uless: "\<forall>x\<in>S \<inter> U - {a}. \<forall>y\<in>S \<inter> U - {a}. d (f x) (f y) < \<epsilon>" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4587 |
using R [OF \<open>\<epsilon>>0\<close>] by blast |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4588 |
then obtain b where b: "b \<noteq> a" "b \<in> S" "b \<in> U" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4589 |
using True by (auto simp add: in_derived_set_of) |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4590 |
have "U \<subseteq> topspace X" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4591 |
by (simp add: U(1) openin_subset) |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4592 |
have "f b \<in> M" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4593 |
using b \<open>openin X U\<close> by (metis image_subset_iff_funcset Int_iff fim image_eqI openin_subset subsetD) |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4594 |
moreover |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4595 |
have "mtopology closure_of f ` ((S \<inter> U) - {a}) \<subseteq> mcball (f b) \<epsilon>" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4596 |
proof (rule closure_of_minimal) |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4597 |
have "f y \<in> M" if "y \<in> S" and "y \<in> U" for y |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4598 |
using \<open>U \<subseteq> topspace X\<close> fim that by (auto simp: Pi_iff) |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4599 |
moreover |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4600 |
have "d (f b) (f y) \<le> \<epsilon>" if "y \<in> S" "y \<in> U" "y \<noteq> a" for y |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4601 |
using that Uless b by force |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4602 |
ultimately show "f ` (S \<inter> U - {a}) \<subseteq> mcball (f b) \<epsilon>" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4603 |
by (force simp: \<open>f b \<in> M\<close>) |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4604 |
qed auto |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4605 |
ultimately show ?thesis |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4606 |
using U by (auto simp add: \<C>_def F_def) |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4607 |
qed |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4608 |
moreover have "\<Inter>\<F> \<noteq> {}" if "finite \<F>" "\<F> \<subseteq> \<C>" for \<F> |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4609 |
proof - |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4610 |
obtain \<G> where sub: "\<G> \<subseteq> {U. openin X U \<and> a \<in> U}" and eq: "\<F> = F ` \<G>" and "finite \<G>" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4611 |
by (metis (no_types, lifting) \<C>_def \<open>\<F> \<subseteq> \<C>\<close> \<open>finite \<F>\<close> finite_subset_image) |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4612 |
then have "U \<subseteq> topspace X" if "U \<in> \<G>" for U |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4613 |
using openin_subset that by auto |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4614 |
then have "T \<subseteq> mtopology closure_of T" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4615 |
if "T \<in> (\<lambda>U. f ` (S \<inter> U - {a})) ` \<G>" for T |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4616 |
using that fim by (fastforce simp add: intro!: closure_of_subset) |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4617 |
moreover |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4618 |
have ain: "a \<in> \<Inter> (insert (topspace X) \<G>)" "openin X (\<Inter> (insert (topspace X) \<G>))" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4619 |
using True in_derived_set_of sub \<open>finite \<G>\<close> by (fastforce intro!: openin_Inter)+ |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4620 |
then obtain y where "y \<noteq> a" "y \<in> S" and y: "y \<in> \<Inter> (insert (topspace X) \<G>)" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4621 |
by (meson \<open>a \<in> X derived_set_of S\<close> sub in_derived_set_of) |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4622 |
then have "f y \<in> \<Inter>\<F>" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4623 |
using eq that ain fim by (auto simp add: F_def image_subset_iff in_closure_of) |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4624 |
then show ?thesis by blast |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4625 |
qed |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4626 |
ultimately have "\<Inter>\<C> \<noteq> {}" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4627 |
using \<open>mcomplete\<close> mcomplete_fip by metis |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4628 |
then obtain b where "b \<in> \<Inter>\<C>" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4629 |
by auto |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4630 |
then have "b \<in> M" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4631 |
using sub_mcball \<C>_clo mbounded_alt_pos mbounded_empty metric_closedin_iff_sequentially_closed by force |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4632 |
have "limitin mtopology f b (atin_within X a S)" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4633 |
proof (clarsimp simp: limitin_metric \<open>b \<in> M\<close>) |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4634 |
fix \<epsilon> :: real |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4635 |
assume "\<epsilon> > 0" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4636 |
then obtain U where U: "openin X U" "a \<in> U" and subU: "U \<subseteq> topspace X" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4637 |
and Uless: "\<forall>x\<in>S \<inter> U - {a}. \<forall>y\<in>S \<inter> U - {a}. d (f x) (f y) < \<epsilon>/2" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4638 |
by (metis R half_gt_zero openin_subset) |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4639 |
then obtain x where x: "x \<in> S" "x \<in> U" "x \<noteq> a" and fx: "f x \<in> mball b (\<epsilon>/2)" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4640 |
using \<open>b \<in> \<Inter>\<C>\<close> |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4641 |
apply (simp add: \<C>_def F_def closure_of_def del: divide_const_simps) |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4642 |
by (metis Diff_iff Int_iff centre_in_mball_iff in_mball openin_mball singletonI zero_less_numeral) |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4643 |
moreover |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4644 |
have "d (f y) b < \<epsilon>" if "y \<in> U" "y \<noteq> a" "y \<in> S" for y |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4645 |
proof - |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4646 |
have "d (f x) (f y) < \<epsilon>/2" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4647 |
using Uless that x by force |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4648 |
moreover have "d b (f x) < \<epsilon>/2" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4649 |
using fx by simp |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4650 |
ultimately show ?thesis |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4651 |
using triangle [of b "f x" "f y"] subU that \<open>b \<in> M\<close> commute fim fx by fastforce |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4652 |
qed |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4653 |
ultimately show "\<forall>\<^sub>F x in atin_within X a S. f x \<in> M \<and> d (f x) b < \<epsilon>" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4654 |
using fim U |
78750
f229090cb8a3
atin now an abbreviation for atin_within, which has been moved to Abstract_Limits
paulson <lp15@cam.ac.uk>
parents:
78336
diff
changeset
|
4655 |
apply (simp add: eventually_atin_within del: divide_const_simps flip: image_subset_iff_funcset) |
78250
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4656 |
by (smt (verit, del_insts) Diff_iff Int_iff imageI insertI1 openin_subset subsetD) |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4657 |
qed |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4658 |
then show ?thesis .. |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4659 |
qed |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4660 |
ultimately |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4661 |
show ?thesis |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4662 |
by (meson True \<open>M \<noteq> {}\<close> in_derived_set_of) |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4663 |
next |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4664 |
case False |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4665 |
have "(\<exists>l. limitin mtopology f l (atin_within X a S))" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4666 |
by (metis \<open>M \<noteq> {}\<close> False derived_set_of_trivial_limit equals0I limitin_trivial topspace_mtopology) |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4667 |
moreover have "\<forall>e>0. \<exists>U. openin X U \<and> a \<in> U \<and> (\<forall>x\<in>S \<inter> U - {a}. \<forall>y\<in>S \<inter> U - {a}. d (f x) (f y) < e)" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4668 |
by (metis Diff_iff False IntE True in_derived_set_of insert_iff) |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4669 |
ultimately show ?thesis |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4670 |
using limitin_mspace by blast |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4671 |
qed |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4672 |
next |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4673 |
case False |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4674 |
then show ?thesis |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4675 |
by (metis derived_set_of_trivial_limit ex_in_conv in_derived_set_of limitin_mspace limitin_trivial topspace_mtopology) |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4676 |
qed |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4677 |
qed |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4678 |
|
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4679 |
text \<open>The HOL Light proof uses some ugly tricks to share common parts of what are two separate proofs for the two cases\<close> |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4680 |
lemma (in Metric_space) gdelta_in_points_of_convergence_within: |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4681 |
assumes "mcomplete" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4682 |
and f: "continuous_map (subtopology X S) mtopology f \<or> t1_space X \<and> f \<in> S \<rightarrow> M" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4683 |
shows "gdelta_in X {x \<in> topspace X. \<exists>l. limitin mtopology f l (atin_within X x S)}" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4684 |
proof - |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4685 |
have fim: "f \<in> (topspace X \<inter> S) \<rightarrow> M" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4686 |
using continuous_map_image_subset_topspace f by force |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4687 |
show ?thesis |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4688 |
proof (cases "M={}") |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4689 |
case True |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4690 |
then show ?thesis |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4691 |
by (smt (verit) Collect_cong empty_def empty_iff gdelta_in_empty limitin_mspace) |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4692 |
next |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4693 |
case False |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4694 |
define A where "A \<equiv> {a \<in> topspace X. \<forall>\<epsilon>>0. \<exists>U. openin X U \<and> a \<in> U \<and> (\<forall>x\<in>S \<inter> U - {a}. \<forall>y\<in>S \<inter> U - {a}. d (f x) (f y) < \<epsilon>)}" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4695 |
have "gdelta_in X A" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4696 |
using f |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4697 |
proof (elim disjE conjE) |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4698 |
assume cm: "continuous_map (subtopology X S) mtopology f" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4699 |
define C where "C \<equiv> \<lambda>r. \<Union>{U. openin X U \<and> (\<forall>x \<in> S\<inter>U. \<forall>y \<in> S\<inter>U. d (f x) (f y) < r)}" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4700 |
define B where "B \<equiv> (\<Inter>n. C(inverse(Suc n)))" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4701 |
define D where "D \<equiv> (\<Inter> (C ` {0<..}))" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4702 |
have "D=B" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4703 |
unfolding B_def C_def D_def |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4704 |
apply (intro Inter_eq_Inter_inverse_Suc Sup_subset_mono) |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4705 |
by (smt (verit, ccfv_threshold) Collect_mono_iff) |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4706 |
have "B \<subseteq> topspace X" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4707 |
using openin_subset by (force simp add: B_def C_def) |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4708 |
have "(countable intersection_of openin X) B" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4709 |
unfolding B_def C_def |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4710 |
by (intro relative_to_inc countable_intersection_of_Inter countable_intersection_of_inc) auto |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4711 |
then have "gdelta_in X B" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4712 |
unfolding gdelta_in_def by (intro relative_to_subset_inc \<open>B \<subseteq> topspace X\<close>) |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4713 |
moreover have "A=D" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4714 |
proof (intro equalityI subsetI) |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4715 |
fix a |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4716 |
assume x: "a \<in> A" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4717 |
then have "a \<in> topspace X" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4718 |
using A_def by blast |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4719 |
show "a \<in> D" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4720 |
proof (clarsimp simp: D_def C_def \<open>a \<in> topspace X\<close>) |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4721 |
fix \<epsilon>::real assume "\<epsilon> > 0" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4722 |
then obtain U where "openin X U" "a \<in> U" and U: "(\<forall>x\<in>S \<inter> U - {a}. \<forall>y\<in>S \<inter> U - {a}. d (f x) (f y) < \<epsilon>)" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4723 |
using x by (force simp: A_def) |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4724 |
show "\<exists>T. openin X T \<and> (\<forall>x\<in>S \<inter> T. \<forall>y\<in>S \<inter> T. d (f x) (f y) < \<epsilon>) \<and> a \<in> T" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4725 |
proof (cases "a \<in> S") |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4726 |
case True |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4727 |
then obtain V where "openin X V" "a \<in> V" and V: "\<forall>x. x \<in> S \<and> x \<in> V \<longrightarrow> f a \<in> M \<and> f x \<in> M \<and> d (f a) (f x) < \<epsilon>" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4728 |
using \<open>a \<in> topspace X\<close> \<open>\<epsilon> > 0\<close> cm |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4729 |
by (force simp add: continuous_map_to_metric openin_subtopology_alt Ball_def) |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4730 |
show ?thesis |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4731 |
proof (intro exI conjI strip) |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4732 |
show "openin X (U \<inter> V)" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4733 |
using \<open>openin X U\<close> \<open>openin X V\<close> by blast |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4734 |
show "a \<in> U \<inter> V" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4735 |
using \<open>a \<in> U\<close> \<open>a \<in> V\<close> by blast |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4736 |
show "\<And>x y. \<lbrakk>x \<in> S \<inter> (U \<inter> V); y \<in> S \<inter> (U \<inter> V)\<rbrakk> \<Longrightarrow> d (f x) (f y) < \<epsilon>" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4737 |
by (metis DiffI Int_iff U V commute singletonD) |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4738 |
qed |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4739 |
next |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4740 |
case False then show ?thesis |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4741 |
using U \<open>a \<in> U\<close> \<open>openin X U\<close> by auto |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4742 |
qed |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4743 |
qed |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4744 |
next |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4745 |
fix x |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4746 |
assume x: "x \<in> D" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4747 |
then have "x \<in> topspace X" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4748 |
using \<open>B \<subseteq> topspace X\<close> \<open>D=B\<close> by blast |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4749 |
with x show "x \<in> A" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4750 |
apply (clarsimp simp: D_def C_def A_def) |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4751 |
by (meson DiffD1 greaterThan_iff) |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4752 |
qed |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4753 |
ultimately show ?thesis |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4754 |
by (simp add: \<open>D=B\<close>) |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4755 |
next |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4756 |
assume "t1_space X" "f \<in> S \<rightarrow> M" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4757 |
define C where "C \<equiv> \<lambda>r. \<Union>{U. openin X U \<and> |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4758 |
(\<exists>b \<in> topspace X. \<forall>x \<in> S\<inter>U - {b}. \<forall>y \<in> S\<inter>U - {b}. d (f x) (f y) < r)}" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4759 |
define B where "B \<equiv> (\<Inter>n. C(inverse(Suc n)))" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4760 |
define D where "D \<equiv> (\<Inter> (C ` {0<..}))" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4761 |
have "D=B" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4762 |
unfolding B_def C_def D_def |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4763 |
apply (intro Inter_eq_Inter_inverse_Suc Sup_subset_mono) |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4764 |
by (smt (verit, ccfv_threshold) Collect_mono_iff) |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4765 |
have "B \<subseteq> topspace X" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4766 |
using openin_subset by (force simp add: B_def C_def) |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4767 |
have "(countable intersection_of openin X) B" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4768 |
unfolding B_def C_def |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4769 |
by (intro relative_to_inc countable_intersection_of_Inter countable_intersection_of_inc) auto |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4770 |
then have "gdelta_in X B" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4771 |
unfolding gdelta_in_def by (intro relative_to_subset_inc \<open>B \<subseteq> topspace X\<close>) |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4772 |
moreover have "A=D" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4773 |
proof (intro equalityI subsetI) |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4774 |
fix x |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4775 |
assume x: "x \<in> D" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4776 |
then have "x \<in> topspace X" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4777 |
using \<open>B \<subseteq> topspace X\<close> \<open>D=B\<close> by blast |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4778 |
show "x \<in> A" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4779 |
proof (clarsimp simp: A_def \<open>x \<in> topspace X\<close>) |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4780 |
fix \<epsilon> :: real |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4781 |
assume "\<epsilon>>0" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4782 |
then obtain U b where "openin X U" "b \<in> topspace X" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4783 |
and U: "\<forall>x\<in>S \<inter> U - {b}. \<forall>y\<in>S \<inter> U - {b}. d (f x) (f y) < \<epsilon>" and "x \<in> U" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4784 |
using x by (auto simp: D_def C_def A_def Ball_def) |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4785 |
then have "openin X (U-{b})" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4786 |
by (meson \<open>t1_space X\<close> t1_space_openin_delete_alt) |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4787 |
then show "\<exists>U. openin X U \<and> x \<in> U \<and> (\<forall>xa\<in>S \<inter> U - {x}. \<forall>y\<in>S \<inter> U - {x}. d (f xa) (f y) < \<epsilon>)" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4788 |
using U \<open>openin X U\<close> \<open>x \<in> U\<close> by auto |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4789 |
qed |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4790 |
next |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4791 |
show "\<And>x. x \<in> A \<Longrightarrow> x \<in> D" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4792 |
unfolding A_def D_def C_def |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4793 |
by clarsimp meson |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4794 |
qed |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4795 |
ultimately show ?thesis |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4796 |
by (simp add: \<open>D=B\<close>) |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4797 |
qed |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4798 |
then show ?thesis |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4799 |
by (simp add: A_def convergent_eq_zero_oscillation_gen False fim \<open>mcomplete\<close> cong: conj_cong) |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4800 |
qed |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4801 |
qed |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4802 |
|
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4803 |
|
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4804 |
lemma gdelta_in_points_of_convergence_within: |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4805 |
assumes Y: "completely_metrizable_space Y" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4806 |
and f: "continuous_map (subtopology X S) Y f \<or> t1_space X \<and> f \<in> S \<rightarrow> topspace Y" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4807 |
shows "gdelta_in X {x \<in> topspace X. \<exists>l. limitin Y f l (atin_within X x S)}" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4808 |
using assms |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4809 |
unfolding completely_metrizable_space_def |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4810 |
using Metric_space.gdelta_in_points_of_convergence_within Metric_space.topspace_mtopology by fastforce |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4811 |
|
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4812 |
|
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4813 |
lemma Lavrentiev_extension_gen: |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4814 |
assumes "S \<subseteq> topspace X" and Y: "completely_metrizable_space Y" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4815 |
and contf: "continuous_map (subtopology X S) Y f" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4816 |
obtains U g where "gdelta_in X U" "S \<subseteq> U" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4817 |
"continuous_map (subtopology X (X closure_of S \<inter> U)) Y g" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4818 |
"\<And>x. x \<in> S \<Longrightarrow> g x = f x" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4819 |
proof - |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4820 |
define U where "U \<equiv> {x \<in> topspace X. \<exists>l. limitin Y f l (atin_within X x S)}" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4821 |
have "S \<subseteq> U" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4822 |
using that contf limit_continuous_map_within subsetD [OF \<open>S \<subseteq> topspace X\<close>] |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4823 |
by (fastforce simp: U_def) |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4824 |
then have "S \<subseteq> X closure_of S \<inter> U" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4825 |
by (simp add: \<open>S \<subseteq> topspace X\<close> closure_of_subset) |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4826 |
moreover |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4827 |
have "\<And>t. t \<in> X closure_of S \<inter> U - S \<Longrightarrow> \<exists>l. limitin Y f l (atin_within X t S)" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4828 |
using U_def by blast |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4829 |
moreover have "regular_space Y" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4830 |
by (simp add: Y completely_metrizable_imp_metrizable_space metrizable_imp_regular_space) |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4831 |
ultimately |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4832 |
obtain g where g: "continuous_map (subtopology X (X closure_of S \<inter> U)) Y g" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4833 |
and gf: "\<And>x. x \<in> S \<Longrightarrow> g x = f x" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4834 |
using continuous_map_extension_pointwise_alt assms by blast |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4835 |
show thesis |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4836 |
proof |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4837 |
show "gdelta_in X U" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4838 |
by (simp add: U_def Y contf gdelta_in_points_of_convergence_within) |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4839 |
show "continuous_map (subtopology X (X closure_of S \<inter> U)) Y g" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4840 |
by (simp add: g) |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4841 |
qed (use \<open>S \<subseteq> U\<close> gf in auto) |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4842 |
qed |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4843 |
|
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4844 |
lemma Lavrentiev_extension: |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4845 |
assumes "S \<subseteq> topspace X" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4846 |
and X: "metrizable_space X \<or> topspace X \<subseteq> X closure_of S" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4847 |
and Y: "completely_metrizable_space Y" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4848 |
and contf: "continuous_map (subtopology X S) Y f" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4849 |
obtains U g where "gdelta_in X U" "S \<subseteq> U" "U \<subseteq> X closure_of S" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4850 |
"continuous_map (subtopology X U) Y g" "\<And>x. x \<in> S \<Longrightarrow> g x = f x" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4851 |
proof - |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4852 |
obtain U g where "gdelta_in X U" "S \<subseteq> U" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4853 |
and contg: "continuous_map (subtopology X (X closure_of S \<inter> U)) Y g" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4854 |
and gf: "\<And>x. x \<in> S \<Longrightarrow> g x = f x" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4855 |
using Lavrentiev_extension_gen Y assms(1) contf by blast |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4856 |
define V where "V \<equiv> X closure_of S \<inter> U" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4857 |
show thesis |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4858 |
proof |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4859 |
show "gdelta_in X V" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4860 |
by (metis V_def X \<open>gdelta_in X U\<close> closed_imp_gdelta_in closedin_closure_of closure_of_subset_topspace gdelta_in_Int gdelta_in_topspace subset_antisym) |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4861 |
show "S \<subseteq> V" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4862 |
by (simp add: V_def \<open>S \<subseteq> U\<close> assms(1) closure_of_subset) |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4863 |
show "continuous_map (subtopology X V) Y g" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4864 |
by (simp add: V_def contg) |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4865 |
qed (auto simp: gf V_def) |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4866 |
qed |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4867 |
|
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4868 |
subsection\<open>Embedding in products and hence more about completely metrizable spaces\<close> |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4869 |
|
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4870 |
lemma (in Metric_space) gdelta_homeomorphic_space_closedin_product: |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4871 |
assumes S: "\<And>i. i \<in> I \<Longrightarrow> openin mtopology (S i)" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4872 |
obtains T where "closedin (prod_topology mtopology (powertop_real I)) T" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4873 |
"subtopology mtopology (\<Inter>i \<in> I. S i) homeomorphic_space |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4874 |
subtopology (prod_topology mtopology (powertop_real I)) T" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4875 |
proof (cases "I={}") |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4876 |
case True |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4877 |
then have top: "topspace (prod_topology mtopology (powertop_real I)) = (M \<times> {(\<lambda>x. undefined)})" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4878 |
by simp |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4879 |
show ?thesis |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4880 |
proof |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4881 |
show "closedin (prod_topology mtopology (powertop_real I)) (M \<times> {(\<lambda>x. undefined)})" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4882 |
by (metis top closedin_topspace) |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4883 |
have "subtopology mtopology (\<Inter>(S ` I)) homeomorphic_space mtopology" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4884 |
by (simp add: True product_topology_empty_discrete) |
78336 | 4885 |
also have "\<dots> homeomorphic_space (prod_topology mtopology (discrete_topology {\<lambda>x. undefined}))" |
4886 |
by (meson homeomorphic_space_sym prod_topology_homeomorphic_space_left) |
|
78250
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4887 |
finally |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4888 |
show "subtopology mtopology (\<Inter>(S ` I)) homeomorphic_space subtopology (prod_topology mtopology (powertop_real I)) (M \<times> {(\<lambda>x. undefined)})" |
78336 | 4889 |
by (smt (verit, ccfv_SIG) True product_topology_empty_discrete subtopology_topspace top) |
78250
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4890 |
qed |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4891 |
next |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4892 |
case False |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4893 |
have SM: "\<And>i. i \<in> I \<Longrightarrow> S i \<subseteq> M" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4894 |
using assms openin_mtopology by blast |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4895 |
then have "(\<Inter>i \<in> I. S i) \<subseteq> M" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4896 |
using False by blast |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4897 |
define dd where "dd \<equiv> \<lambda>i. if i\<notin>I \<or> S i = M then \<lambda>u. 1 else (\<lambda>u. INF x \<in> M - S i. d u x)" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4898 |
have [simp]: "bdd_below (d u ` A)" for u A |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4899 |
by (meson bdd_belowI2 nonneg) |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4900 |
have cont_dd: "continuous_map (subtopology mtopology (S i)) euclidean (dd i)" if "i \<in> I" for i |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4901 |
proof - |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4902 |
have "dist (Inf (d x ` (M - S i))) (Inf (d y ` (M - S i))) \<le> d x y" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4903 |
if "x \<in> S i" "x \<in> M" "y \<in> S i" "y \<in> M" "S i \<noteq> M" for x y |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4904 |
proof - |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4905 |
have [simp]: "\<not> M \<subseteq> S i" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4906 |
using SM \<open>S i \<noteq> M\<close> \<open>i \<in> I\<close> by auto |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4907 |
have "\<And>u. \<lbrakk>u \<in> M; u \<notin> S i\<rbrakk> \<Longrightarrow> Inf (d x ` (M - S i)) \<le> d x y + d y u" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4908 |
apply (clarsimp simp add: cInf_le_iff_less) |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4909 |
by (smt (verit) DiffI triangle \<open>x \<in> M\<close> \<open>y \<in> M\<close>) |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4910 |
then have "Inf (d x ` (M - S i)) - d x y \<le> Inf (d y ` (M - S i))" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4911 |
by (force simp add: le_cInf_iff) |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4912 |
moreover |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4913 |
have "\<And>u. \<lbrakk>u \<in> M; u \<notin> S i\<rbrakk> \<Longrightarrow> Inf (d y ` (M - S i)) \<le> d x u + d x y" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4914 |
apply (clarsimp simp add: cInf_le_iff_less) |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4915 |
by (smt (verit) DiffI triangle'' \<open>x \<in> M\<close> \<open>y \<in> M\<close>) |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4916 |
then have "Inf (d y ` (M - S i)) - d x y \<le> Inf (d x ` (M - S i))" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4917 |
by (force simp add: le_cInf_iff) |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4918 |
ultimately show ?thesis |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4919 |
by (simp add: dist_real_def abs_le_iff) |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4920 |
qed |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4921 |
then have *: "Lipschitz_continuous_map (submetric Self (S i)) euclidean_metric (\<lambda>u. Inf (d u ` (M - S i)))" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4922 |
unfolding Lipschitz_continuous_map_def by (force intro!: exI [where x=1]) |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4923 |
then show ?thesis |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4924 |
using Lipschitz_continuous_imp_continuous_map [OF *] |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4925 |
by (simp add: dd_def Self_def mtopology_of_submetric ) |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4926 |
qed |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4927 |
have dd_pos: "0 < dd i x" if "x \<in> S i" for i x |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4928 |
proof (clarsimp simp add: dd_def) |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4929 |
assume "i \<in> I" and "S i \<noteq> M" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4930 |
have opeS: "openin mtopology (S i)" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4931 |
by (simp add: \<open>i \<in> I\<close> assms) |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4932 |
then obtain r where "r>0" and r: "\<And>y. \<lbrakk>y \<in> M; d x y < r\<rbrakk> \<Longrightarrow> y \<in> S i" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4933 |
by (meson \<open>x \<in> S i\<close> in_mball openin_mtopology subsetD) |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4934 |
then have "\<And>y. y \<in> M - S i \<Longrightarrow> d x y \<ge> r" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4935 |
by (meson Diff_iff linorder_not_le) |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4936 |
then have "Inf (d x ` (M - S i)) \<ge> r" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4937 |
by (meson Diff_eq_empty_iff SM \<open>S i \<noteq> M\<close> \<open>i \<in> I\<close> cINF_greatest set_eq_subset) |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4938 |
with \<open>r>0\<close> show "0 < Inf (d x ` (M - S i))" by simp |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4939 |
qed |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4940 |
define f where "f \<equiv> \<lambda>x. (x, \<lambda>i\<in>I. inverse(dd i x))" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4941 |
define T where "T \<equiv> f ` (\<Inter>i \<in> I. S i)" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4942 |
show ?thesis |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4943 |
proof |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4944 |
show "closedin (prod_topology mtopology (powertop_real I)) T" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4945 |
unfolding closure_of_subset_eq [symmetric] |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4946 |
proof |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4947 |
show "T \<subseteq> topspace (prod_topology mtopology (powertop_real I))" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4948 |
using False SM by (auto simp: T_def f_def) |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4949 |
|
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4950 |
have "(x,ds) \<in> T" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4951 |
if \<section>: "\<And>U. \<lbrakk>(x, ds) \<in> U; openin (prod_topology mtopology (powertop_real I)) U\<rbrakk> \<Longrightarrow> \<exists>y\<in>T. y \<in> U" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4952 |
and "x \<in> M" and ds: "ds \<in> I \<rightarrow>\<^sub>E UNIV" for x ds |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4953 |
proof - |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4954 |
have ope: "\<exists>x. x \<in> \<Inter>(S ` I) \<and> f x \<in> U \<times> V" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4955 |
if "x \<in> U" and "ds \<in> V" and "openin mtopology U" and "openin (powertop_real I) V" for U V |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4956 |
using \<section> [of "U\<times>V"] that by (force simp add: T_def openin_prod_Times_iff) |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4957 |
have x_in_INT: "x \<in> \<Inter>(S ` I)" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4958 |
proof clarify |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4959 |
fix i |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4960 |
assume "i \<in> I" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4961 |
show "x \<in> S i" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4962 |
proof (rule ccontr) |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4963 |
assume "x \<notin> S i" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4964 |
have "openin (powertop_real I) {z \<in> topspace (powertop_real I). z i \<in> {ds i - 1 <..< ds i + 1}}" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4965 |
proof (rule openin_continuous_map_preimage) |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4966 |
show "continuous_map (powertop_real I) euclidean (\<lambda>x. x i)" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4967 |
by (metis \<open>i \<in> I\<close> continuous_map_product_projection) |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4968 |
qed auto |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4969 |
then obtain y where "y \<in> S i" "y \<in> M" and dxy: "d x y < inverse (\<bar>ds i\<bar> + 1)" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4970 |
and intvl: "inverse (dd i y) \<in> {ds i - 1 <..< ds i + 1}" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4971 |
using ope [of "mball x (inverse(abs(ds i) + 1))" "{z \<in> topspace(powertop_real I). z i \<in> {ds i - 1<..<ds i + 1}}"] |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4972 |
\<open>x \<in> M\<close> \<open>ds \<in> I \<rightarrow>\<^sub>E UNIV\<close> \<open>i \<in> I\<close> |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4973 |
by (fastforce simp add: f_def) |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4974 |
have "\<not> M \<subseteq> S i" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4975 |
using \<open>x \<notin> S i\<close> \<open>x \<in> M\<close> by blast |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4976 |
have "inverse (\<bar>ds i\<bar> + 1) \<le> dd i y" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4977 |
using intvl \<open>y \<in> S i\<close> dd_pos [of y i] |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4978 |
by (smt (verit, ccfv_threshold) greaterThanLessThan_iff inverse_inverse_eq le_imp_inverse_le) |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4979 |
also have "\<dots> \<le> d x y" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4980 |
using \<open>i \<in> I\<close> \<open>\<not> M \<subseteq> S i\<close> \<open>x \<notin> S i\<close> \<open>x \<in> M\<close> |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4981 |
apply (simp add: dd_def cInf_le_iff_less) |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4982 |
using commute by force |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4983 |
finally show False |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4984 |
using dxy by linarith |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4985 |
qed |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4986 |
qed |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4987 |
moreover have "ds = (\<lambda>i\<in>I. inverse (dd i x))" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4988 |
proof (rule PiE_ext [OF ds]) |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4989 |
fix i |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4990 |
assume "i \<in> I" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4991 |
define e where "e \<equiv> \<bar>ds i - inverse (dd i x)\<bar>" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4992 |
{ assume con: "e > 0" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4993 |
have "continuous_map (subtopology mtopology (S i)) euclidean (\<lambda>x. inverse (dd i x))" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4994 |
using dd_pos cont_dd \<open>i \<in> I\<close> |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4995 |
by (fastforce simp: intro!: continuous_map_real_inverse) |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4996 |
then have "openin (subtopology mtopology (S i)) |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4997 |
{z \<in> topspace (subtopology mtopology (S i)). |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4998 |
inverse (dd i z) \<in> {inverse(dd i x) - e/2<..<inverse(dd i x) + e/2}}" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
4999 |
using openin_continuous_map_preimage open_greaterThanLessThan open_openin by blast |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
5000 |
then obtain U where "openin mtopology U" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
5001 |
and U: "{z \<in> S i. inverse (dd i x) - e/2 < inverse (dd i z) \<and> |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
5002 |
inverse (dd i z) < inverse (dd i x) + e/2} |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
5003 |
= U \<inter> S i" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
5004 |
using SM \<open>i \<in> I\<close> by (auto simp: openin_subtopology) |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
5005 |
have "x \<in> U" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
5006 |
using U x_in_INT \<open>i \<in> I\<close> con by fastforce |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
5007 |
have "ds \<in> {z \<in> topspace (powertop_real I). z i \<in> {ds i - e / 2<..<ds i + e/2}}" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
5008 |
by (simp add: con ds) |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
5009 |
moreover |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
5010 |
have "openin (powertop_real I) {z \<in> topspace (powertop_real I). z i \<in> {ds i - e / 2<..<ds i + e/2}}" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
5011 |
proof (rule openin_continuous_map_preimage) |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
5012 |
show "continuous_map (powertop_real I) euclidean (\<lambda>x. x i)" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
5013 |
by (metis \<open>i \<in> I\<close> continuous_map_product_projection) |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
5014 |
qed auto |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
5015 |
ultimately obtain y where "y \<in> \<Inter>(S ` I) \<and> f y \<in> U \<times> {z \<in> topspace (powertop_real I). z i \<in> {ds i - e / 2<..<ds i + e/2}}" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
5016 |
using ope \<open>x \<in> U\<close> \<open>openin mtopology U\<close> \<open>x \<in> U\<close> |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
5017 |
by presburger |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
5018 |
with \<open>i \<in> I\<close> U |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
5019 |
have False unfolding set_eq_iff f_def e_def by simp (smt (verit) field_sum_of_halves) |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
5020 |
} |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
5021 |
then show "ds i = (\<lambda>i\<in>I. inverse (dd i x)) i" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
5022 |
using \<open>i \<in> I\<close> by (force simp: e_def) |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
5023 |
qed auto |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
5024 |
ultimately show ?thesis |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
5025 |
by (auto simp: T_def f_def) |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
5026 |
qed |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
5027 |
then show "prod_topology mtopology (powertop_real I) closure_of T \<subseteq> T" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
5028 |
by (auto simp: closure_of_def) |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
5029 |
qed |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
5030 |
have eq: "(\<Inter>(S ` I) \<times> (I \<rightarrow>\<^sub>E UNIV) \<inter> f ` (M \<inter> \<Inter>(S ` I))) = (f ` \<Inter>(S ` I))" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
5031 |
using False SM by (force simp: f_def image_iff) |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
5032 |
have "continuous_map (subtopology mtopology (\<Inter>(S ` I))) euclidean (dd i)" if "i \<in> I" for i |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
5033 |
by (meson INT_lower cont_dd continuous_map_from_subtopology_mono that) |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
5034 |
then have "continuous_map (subtopology mtopology (\<Inter>(S ` I))) (powertop_real I) (\<lambda>x. \<lambda>i\<in>I. inverse (dd i x))" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
5035 |
using dd_pos by (fastforce simp: continuous_map_componentwise intro!: continuous_map_real_inverse) |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
5036 |
then have "embedding_map (subtopology mtopology (\<Inter>(S ` I))) (prod_topology (subtopology mtopology (\<Inter>(S ` I))) (powertop_real I)) f" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
5037 |
by (simp add: embedding_map_graph f_def) |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
5038 |
moreover have "subtopology (prod_topology (subtopology mtopology (\<Inter>(S ` I))) (powertop_real I)) |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
5039 |
(f ` topspace (subtopology mtopology (\<Inter>(S ` I)))) = |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
5040 |
subtopology (prod_topology mtopology (powertop_real I)) T" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
5041 |
by (simp add: prod_topology_subtopology subtopology_subtopology T_def eq) |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
5042 |
ultimately |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
5043 |
show "subtopology mtopology (\<Inter>(S ` I)) homeomorphic_space subtopology (prod_topology mtopology (powertop_real I)) T" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
5044 |
by (metis embedding_map_imp_homeomorphic_space) |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
5045 |
qed |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
5046 |
qed |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
5047 |
|
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
5048 |
|
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
5049 |
lemma gdelta_homeomorphic_space_closedin_product: |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
5050 |
assumes "metrizable_space X" and "\<And>i. i \<in> I \<Longrightarrow> openin X (S i)" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
5051 |
obtains T where "closedin (prod_topology X (powertop_real I)) T" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
5052 |
"subtopology X (\<Inter>i \<in> I. S i) homeomorphic_space |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
5053 |
subtopology (prod_topology X (powertop_real I)) T" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
5054 |
using Metric_space.gdelta_homeomorphic_space_closedin_product |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
5055 |
by (metis assms metrizable_space_def) |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
5056 |
|
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
5057 |
lemma open_homeomorphic_space_closedin_product: |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
5058 |
assumes "metrizable_space X" and "openin X S" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
5059 |
obtains T where "closedin (prod_topology X euclideanreal) T" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
5060 |
"subtopology X S homeomorphic_space |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
5061 |
subtopology (prod_topology X euclideanreal) T" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
5062 |
proof - |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
5063 |
obtain T where cloT: "closedin (prod_topology X (powertop_real {()})) T" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
5064 |
and homT: "subtopology X S homeomorphic_space |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
5065 |
subtopology (prod_topology X (powertop_real {()})) T" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
5066 |
using gdelta_homeomorphic_space_closedin_product [of X "{()}" "\<lambda>i. S"] assms |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
5067 |
by auto |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
5068 |
have "prod_topology X (powertop_real {()}) homeomorphic_space prod_topology X euclideanreal" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
5069 |
by (meson homeomorphic_space_prod_topology homeomorphic_space_refl homeomorphic_space_singleton_product) |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
5070 |
then obtain f where f: "homeomorphic_map (prod_topology X (powertop_real {()})) (prod_topology X euclideanreal) f" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
5071 |
unfolding homeomorphic_space by metis |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
5072 |
show thesis |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
5073 |
proof |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
5074 |
show "closedin (prod_topology X euclideanreal) (f ` T)" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
5075 |
using cloT f homeomorphic_map_closedness_eq by blast |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
5076 |
moreover have "T = topspace (subtopology (prod_topology X (powertop_real {()})) T)" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
5077 |
by (metis cloT closedin_subset topspace_subtopology_subset) |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
5078 |
ultimately show "subtopology X S homeomorphic_space subtopology (prod_topology X euclideanreal) (f ` T)" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
5079 |
by (smt (verit, best) closedin_subset f homT homeomorphic_map_subtopologies homeomorphic_space |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
5080 |
homeomorphic_space_trans topspace_subtopology topspace_subtopology_subset) |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
5081 |
qed |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
5082 |
qed |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
5083 |
|
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
5084 |
lemma completely_metrizable_space_gdelta_in_alt: |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
5085 |
assumes X: "completely_metrizable_space X" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
5086 |
and S: "(countable intersection_of openin X) S" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
5087 |
shows "completely_metrizable_space (subtopology X S)" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
5088 |
proof - |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
5089 |
obtain \<U> where "countable \<U>" "S = \<Inter>\<U>" and ope: "\<And>U. U \<in> \<U> \<Longrightarrow> openin X U" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
5090 |
using S by (force simp add: intersection_of_def) |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
5091 |
then have \<U>: "completely_metrizable_space (powertop_real \<U>)" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
5092 |
by (simp add: completely_metrizable_space_euclidean completely_metrizable_space_product_topology) |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
5093 |
obtain C where "closedin (prod_topology X (powertop_real \<U>)) C" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
5094 |
and sub: "subtopology X (\<Inter>\<U>) homeomorphic_space |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
5095 |
subtopology (prod_topology X (powertop_real \<U>)) C" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
5096 |
by (metis gdelta_homeomorphic_space_closedin_product X completely_metrizable_imp_metrizable_space ope INF_identity_eq) |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
5097 |
moreover have "completely_metrizable_space (prod_topology X (powertop_real \<U>))" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
5098 |
by (simp add: completely_metrizable_space_prod_topology X \<U>) |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
5099 |
ultimately have "completely_metrizable_space (subtopology (prod_topology X (powertop_real \<U>)) C)" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
5100 |
using completely_metrizable_space_closedin by blast |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
5101 |
then show ?thesis |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
5102 |
using \<open>S = \<Inter>\<U>\<close> sub homeomorphic_completely_metrizable_space by blast |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
5103 |
qed |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
5104 |
|
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
5105 |
lemma completely_metrizable_space_gdelta_in: |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
5106 |
"\<lbrakk>completely_metrizable_space X; gdelta_in X S\<rbrakk> |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
5107 |
\<Longrightarrow> completely_metrizable_space (subtopology X S)" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
5108 |
by (simp add: completely_metrizable_space_gdelta_in_alt gdelta_in_alt) |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
5109 |
|
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
5110 |
lemma completely_metrizable_space_openin: |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
5111 |
"\<lbrakk>completely_metrizable_space X; openin X S\<rbrakk> |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
5112 |
\<Longrightarrow> completely_metrizable_space (subtopology X S)" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
5113 |
by (simp add: completely_metrizable_space_gdelta_in open_imp_gdelta_in) |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
5114 |
|
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
5115 |
|
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
5116 |
lemma (in Metric_space) locally_compact_imp_completely_metrizable_space: |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
5117 |
assumes "locally_compact_space mtopology" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
5118 |
shows "completely_metrizable_space mtopology" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
5119 |
proof - |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
5120 |
obtain f :: "['a,'a] \<Rightarrow> real" and m' where |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
5121 |
"mcomplete_of m'" and fim: "f \<in> M \<rightarrow> mspace m'" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
5122 |
and clo: "mtopology_of m' closure_of f ` M = mspace m'" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
5123 |
and d: "\<And>x y. \<lbrakk>x \<in> M; y \<in> M\<rbrakk> \<Longrightarrow> mdist m' (f x) (f y) = d x y" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
5124 |
by (metis metric_completion) |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
5125 |
then have "embedding_map mtopology (mtopology_of m') f" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
5126 |
unfolding mtopology_of_def |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
5127 |
by (metis Metric_space12.isometry_imp_embedding_map Metric_space12_mspace_mdist mdist_metric mspace_metric) |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
5128 |
then have hom: "mtopology homeomorphic_space subtopology (mtopology_of m') (f ` M)" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
5129 |
by (metis embedding_map_imp_homeomorphic_space topspace_mtopology) |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
5130 |
have "locally_compact_space (subtopology (mtopology_of m') (f ` M))" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
5131 |
using assms hom homeomorphic_locally_compact_space by blast |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
5132 |
moreover have "Hausdorff_space (mtopology_of m')" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
5133 |
by (simp add: Metric_space.Hausdorff_space_mtopology mtopology_of_def) |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
5134 |
ultimately have "openin (mtopology_of m') (f ` M)" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
5135 |
by (simp add: clo dense_locally_compact_openin_Hausdorff_space fim image_subset_iff_funcset) |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
5136 |
then |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
5137 |
have "completely_metrizable_space (subtopology (mtopology_of m') (f ` M))" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
5138 |
using \<open>mcomplete_of m'\<close> unfolding mcomplete_of_def mtopology_of_def |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
5139 |
by (metis Metric_space.completely_metrizable_space_mtopology Metric_space_mspace_mdist completely_metrizable_space_openin ) |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
5140 |
then show ?thesis |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
5141 |
using hom homeomorphic_completely_metrizable_space by blast |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
5142 |
qed |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
5143 |
|
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
5144 |
lemma locally_compact_imp_completely_metrizable_space: |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
5145 |
assumes "metrizable_space X" and "locally_compact_space X" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
5146 |
shows "completely_metrizable_space X" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
5147 |
by (metis Metric_space.locally_compact_imp_completely_metrizable_space assms metrizable_space_def) |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
5148 |
|
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
5149 |
|
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
5150 |
lemma completely_metrizable_space_imp_gdelta_in: |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
5151 |
assumes X: "metrizable_space X" and "S \<subseteq> topspace X" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
5152 |
and XS: "completely_metrizable_space (subtopology X S)" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
5153 |
shows "gdelta_in X S" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
5154 |
proof - |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
5155 |
obtain U f where "gdelta_in X U" "S \<subseteq> U" and U: "U \<subseteq> X closure_of S" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
5156 |
and contf: "continuous_map (subtopology X U) (subtopology X S) f" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
5157 |
and fid: "\<And>x. x \<in> S \<Longrightarrow> f x = x" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
5158 |
using Lavrentiev_extension[of S X "subtopology X S" id] assms by auto |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
5159 |
then have "f ` topspace (subtopology X U) \<subseteq> topspace (subtopology X S)" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
5160 |
using continuous_map_image_subset_topspace by blast |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
5161 |
then have "f`U \<subseteq> S" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
5162 |
by (metis \<open>gdelta_in X U\<close> \<open>S \<subseteq> topspace X\<close> gdelta_in_subset topspace_subtopology_subset) |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
5163 |
moreover |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
5164 |
have "Hausdorff_space (subtopology X U)" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
5165 |
by (simp add: Hausdorff_space_subtopology X metrizable_imp_Hausdorff_space) |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
5166 |
then have "\<And>x. x \<in> U \<Longrightarrow> f x = x" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
5167 |
using U fid contf forall_in_closure_of_eq [of _ "subtopology X U" S "subtopology X U" f id] |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
5168 |
by (metis \<open>S \<subseteq> U\<close> closure_of_subtopology_open continuous_map_id continuous_map_in_subtopology id_apply inf.orderE subtopology_subtopology) |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
5169 |
ultimately have "U \<subseteq> S" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
5170 |
by auto |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
5171 |
then show ?thesis |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
5172 |
using \<open>S \<subseteq> U\<close> \<open>gdelta_in X U\<close> by auto |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
5173 |
qed |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
5174 |
|
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
5175 |
lemma completely_metrizable_space_eq_gdelta_in: |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
5176 |
"\<lbrakk>completely_metrizable_space X; S \<subseteq> topspace X\<rbrakk> |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
5177 |
\<Longrightarrow> completely_metrizable_space (subtopology X S) \<longleftrightarrow> gdelta_in X S" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
5178 |
using completely_metrizable_imp_metrizable_space completely_metrizable_space_gdelta_in completely_metrizable_space_imp_gdelta_in by blast |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
5179 |
|
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
5180 |
lemma gdelta_in_eq_completely_metrizable_space: |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
5181 |
"completely_metrizable_space X |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
5182 |
\<Longrightarrow> gdelta_in X S \<longleftrightarrow> S \<subseteq> topspace X \<and> completely_metrizable_space (subtopology X S)" |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
5183 |
by (metis completely_metrizable_space_eq_gdelta_in gdelta_in_alt) |
400aecdfd71f
Another tranche of HOL Light material on metric and topological spaces
paulson <lp15@cam.ac.uk>
parents:
78248
diff
changeset
|
5184 |
|
78274
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5185 |
subsection\<open> Theorems from Kuratowski\<close> |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5186 |
|
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5187 |
text\<open>Kuratowski, Remark on an Invariance Theorem, \emph{Fundamenta Mathematicae} \textbf{37} (1950), pp. 251-252. |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5188 |
The idea is that in suitable spaces, to show "number of components of the complement" (without |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5189 |
distinguishing orders of infinity) is a homeomorphic invariant, it |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5190 |
suffices to show it for closed subsets. Kuratowski states the main result |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5191 |
for a "locally connected continuum", and seems clearly to be implicitly |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5192 |
assuming that means metrizable. We call out the general topological |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5193 |
hypotheses more explicitly, which do not however include connectedness. \<close> |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5194 |
|
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5195 |
lemma separation_by_closed_intermediates_count: |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5196 |
assumes X: "hereditarily normal_space X" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5197 |
and "finite \<U>" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5198 |
and pwU: "pairwise (separatedin X) \<U>" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5199 |
and nonempty: "{} \<notin> \<U>" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5200 |
and UU: "\<Union>\<U> = topspace X - S" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5201 |
obtains C where "closedin X C" "C \<subseteq> S" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5202 |
"\<And>D. \<lbrakk>closedin X D; C \<subseteq> D; D \<subseteq> S\<rbrakk> |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5203 |
\<Longrightarrow> \<exists>\<V>. \<V> \<approx> \<U> \<and> pairwise (separatedin X) \<V> \<and> {} \<notin> \<V> \<and> \<Union>\<V> = topspace X - D" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5204 |
proof - |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5205 |
obtain F where F: "\<And>S. S \<in> \<U> \<Longrightarrow> openin X (F S) \<and> S \<subseteq> F S" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5206 |
and pwF: "pairwise (\<lambda>S T. disjnt (F S) (F T)) \<U>" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5207 |
using assms by (smt (verit, best) Diff_subset Sup_le_iff hereditarily_normal_separation_pairwise) |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5208 |
show thesis |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5209 |
proof |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5210 |
show "closedin X (topspace X - \<Union>(F ` \<U>))" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5211 |
using F by blast |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5212 |
show "topspace X - \<Union>(F ` \<U>) \<subseteq> S" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5213 |
using UU F by auto |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5214 |
show "\<exists>\<V>. \<V> \<approx> \<U> \<and> pairwise (separatedin X) \<V> \<and> {} \<notin> \<V> \<and> \<Union>\<V> = topspace X - C" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5215 |
if "closedin X C" "C \<subseteq> S" and C: "topspace X - \<Union>(F ` \<U>) \<subseteq> C" for C |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5216 |
proof (intro exI conjI strip) |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5217 |
have "inj_on (\<lambda>S. F S - C) \<U>" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5218 |
using pwF F |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5219 |
unfolding inj_on_def pairwise_def disjnt_iff |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5220 |
by (metis Diff_iff UU UnionI nonempty subset_empty subset_eq \<open>C \<subseteq> S\<close>) |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5221 |
then show "(\<lambda>S. F S - C) ` \<U> \<approx> \<U>" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5222 |
by simp |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5223 |
show "pairwise (separatedin X) ((\<lambda>S. F S - C) ` \<U>)" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5224 |
using \<open>closedin X C\<close> F pwF by (force simp: pairwise_def openin_diff separatedin_open_sets disjnt_iff) |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5225 |
show "{} \<notin> (\<lambda>S. F S - C) ` \<U>" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5226 |
using nonempty UU \<open>C \<subseteq> S\<close> F |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5227 |
by clarify (metis DiffD2 Diff_eq_empty_iff F UnionI subset_empty subset_eq) |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5228 |
show "(\<Union>S\<in>\<U>. F S - C) = topspace X - C" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5229 |
using UU F C openin_subset by fastforce |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5230 |
qed |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5231 |
qed |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5232 |
qed |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5233 |
|
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5234 |
lemma separation_by_closed_intermediates_gen: |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5235 |
assumes X: "hereditarily normal_space X" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5236 |
and discon: "\<not> connectedin X (topspace X - S)" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5237 |
obtains C where "closedin X C" "C \<subseteq> S" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5238 |
"\<And>D. \<lbrakk>closedin X D; C \<subseteq> D; D \<subseteq> S\<rbrakk> \<Longrightarrow> \<not> connectedin X (topspace X - D)" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5239 |
proof - |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5240 |
obtain C1 C2 where Ueq: "C1 \<union> C2 = topspace X - S" and "C1 \<noteq> {}" "C2 \<noteq> {}" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5241 |
and sep: "separatedin X C1 C2" and "C1 \<noteq> C2" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5242 |
by (metis Diff_subset connectedin_eq_not_separated discon separatedin_refl) |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5243 |
then obtain C where "closedin X C" "C \<subseteq> S" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5244 |
and C: "\<And>D. \<lbrakk>closedin X D; C \<subseteq> D; D \<subseteq> S\<rbrakk> |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5245 |
\<Longrightarrow> \<exists>\<V>. \<V> \<approx> {C1,C2} \<and> pairwise (separatedin X) \<V> \<and> {} \<notin> \<V> \<and> \<Union>\<V> = topspace X - D" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5246 |
using separation_by_closed_intermediates_count [of X "{C1,C2}" S] X |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5247 |
apply (simp add: pairwise_insert separatedin_sym) |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5248 |
by metis |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5249 |
have "\<not> connectedin X (topspace X - D)" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5250 |
if D: "closedin X D" "C \<subseteq> D" "D \<subseteq> S" for D |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5251 |
proof - |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5252 |
obtain V1 V2 where *: "pairwise (separatedin X) {V1,V2}" "{} \<notin> {V1,V2}" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5253 |
"\<Union>{V1,V2} = topspace X - D" "V1\<noteq>V2" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5254 |
by (metis C [OF D] \<open>C1 \<noteq> C2\<close> eqpoll_doubleton_iff) |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5255 |
then have "disjnt V1 V2" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5256 |
by (metis pairwise_insert separatedin_imp_disjoint singleton_iff) |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5257 |
with * show ?thesis |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5258 |
by (auto simp add: connectedin_eq_not_separated pairwise_insert) |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5259 |
qed |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5260 |
then show thesis |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5261 |
using \<open>C \<subseteq> S\<close> \<open>closedin X C\<close> that by auto |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5262 |
qed |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5263 |
|
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5264 |
lemma separation_by_closed_intermediates_eq_count: |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5265 |
fixes n::nat |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5266 |
assumes lcX: "locally_connected_space X" and hnX: "hereditarily normal_space X" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5267 |
shows "(\<exists>\<U>. \<U> \<approx> {..<n} \<and> pairwise (separatedin X) \<U> \<and> {} \<notin> \<U> \<and> \<Union>\<U> = topspace X - S) \<longleftrightarrow> |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5268 |
(\<exists>C. closedin X C \<and> C \<subseteq> S \<and> |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5269 |
(\<forall>D. closedin X D \<and> C \<subseteq> D \<and> D \<subseteq> S |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5270 |
\<longrightarrow> (\<exists>\<U>. \<U> \<approx> {..<n} \<and> pairwise (separatedin X) \<U> \<and> {} \<notin> \<U> \<and> \<Union>\<U> = topspace X - D)))" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5271 |
(is "?lhs = ?rhs") |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5272 |
proof |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5273 |
assume ?lhs then show ?rhs |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5274 |
by (smt (verit, best) hnX separation_by_closed_intermediates_count eqpoll_iff_finite_card eqpoll_trans) |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5275 |
next |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5276 |
assume R: ?rhs |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5277 |
show ?lhs |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5278 |
proof (cases "n=0") |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5279 |
case True |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5280 |
with R show ?thesis |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5281 |
by fastforce |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5282 |
next |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5283 |
case False |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5284 |
obtain C where "closedin X C" "C \<subseteq> S" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5285 |
and C: "\<And>D. \<lbrakk>closedin X D; C \<subseteq> D; D \<subseteq> S\<rbrakk> |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5286 |
\<Longrightarrow> \<exists>\<U>. \<U> \<approx> {..<n} \<and> pairwise (separatedin X) \<U> \<and> {} \<notin> \<U> \<and> \<Union>\<U> = topspace X - D" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5287 |
using R by force |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5288 |
then have "C \<subseteq> topspace X" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5289 |
by (simp add: closedin_subset) |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5290 |
define \<U> where "\<U> \<equiv> {D \<in> connected_components_of (subtopology X (topspace X - C)). D-S \<noteq> {}}" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5291 |
have ope\<U>: "openin X U" if "U \<in> \<U>" for U |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5292 |
using that \<open>closedin X C\<close> lcX locally_connected_space_open_connected_components |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5293 |
by (fastforce simp add: closedin_def \<U>_def) |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5294 |
have "{} \<notin> \<U>" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5295 |
by (auto simp: \<U>_def) |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5296 |
have "pairwise disjnt \<U>" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5297 |
using connected_components_of_disjoint by (fastforce simp add: pairwise_def \<U>_def) |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5298 |
show ?lhs |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5299 |
proof (rule ccontr) |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5300 |
assume con: "\<nexists>\<U>. \<U> \<approx> {..<n} \<and> pairwise (separatedin X) \<U> \<and> {} \<notin> \<U> \<and> \<Union>\<U> = topspace X - S" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5301 |
have card\<U>: "finite \<U> \<and> card \<U> < n" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5302 |
proof (rule ccontr) |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5303 |
assume "\<not> (finite \<U> \<and> card \<U> < n)" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5304 |
then obtain \<V> where "\<V> \<subseteq> \<U>" "finite \<V>" "card \<V> = n" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5305 |
by (metis infinite_arbitrarily_large linorder_not_less obtain_subset_with_card_n) |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5306 |
then obtain T where "T \<in> \<V>" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5307 |
using False by force |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5308 |
define \<W> where "\<W> \<equiv> insert (topspace X - S - \<Union>(\<V> - {T})) ((\<lambda>D. D - S) ` (\<V> - {T}))" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5309 |
have "\<Union>\<W> = topspace X - S" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5310 |
using \<open>\<And>U. U \<in> \<U> \<Longrightarrow> openin X U\<close> \<open>\<V> \<subseteq> \<U>\<close> topspace_def by (fastforce simp: \<W>_def) |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5311 |
moreover have "{} \<notin> \<W>" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5312 |
proof - |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5313 |
obtain a where "a \<in> T" "a \<notin> S" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5314 |
using \<U>_def \<open>T \<in> \<V>\<close> \<open>\<V> \<subseteq> \<U>\<close> by blast |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5315 |
then have "a \<in> topspace X" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5316 |
using \<open>T \<in> \<V>\<close> ope\<U> \<open>\<V> \<subseteq> \<U>\<close> openin_subset by blast |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5317 |
moreover have "a \<notin> \<Union>(\<V> - {T})" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5318 |
using diff_Union_pairwise_disjoint [of \<V> "{T}"] \<open>disjoint \<U>\<close> pairwise_subset \<open>T \<in> \<V>\<close> \<open>\<V> \<subseteq> \<U>\<close> \<open>a \<in> T\<close> |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5319 |
by auto |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5320 |
ultimately have "topspace X - S - \<Union>(\<V> - {T}) \<noteq> {}" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5321 |
using \<open>a \<notin> S\<close> by blast |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5322 |
moreover have "\<And>V. V \<in> \<V> - {T} \<Longrightarrow> V - S \<noteq> {}" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5323 |
using \<U>_def \<open>\<V> \<subseteq> \<U>\<close> by blast |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5324 |
ultimately show ?thesis |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5325 |
by (metis (no_types, lifting) \<W>_def image_iff insert_iff) |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5326 |
qed |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5327 |
moreover have "disjoint \<V>" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5328 |
using \<open>\<V> \<subseteq> \<U>\<close> \<open>disjoint \<U>\<close> pairwise_subset by blast |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5329 |
then have inj: "inj_on (\<lambda>D. D - S) (\<V> - {T})" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5330 |
unfolding inj_on_def using \<open>\<V> \<subseteq> \<U>\<close> disjointD \<U>_def inf_commute by blast |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5331 |
have "finite \<W>" "card \<W> = n" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5332 |
using \<open>{} \<notin> \<W>\<close> \<open>n \<noteq> 0\<close> \<open>T \<in> \<V>\<close> |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5333 |
by (auto simp add: \<W>_def \<open>finite \<V>\<close> card_insert_if card_image inj \<open>card \<V> = n\<close>) |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5334 |
moreover have "pairwise (separatedin X) \<W>" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5335 |
proof - |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5336 |
have "disjoint \<W>" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5337 |
using \<open>disjoint \<V>\<close> by (auto simp: \<W>_def pairwise_def disjnt_iff) |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5338 |
have "pairwise (separatedin (subtopology X (topspace X - S))) \<W>" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5339 |
proof (intro pairwiseI) |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5340 |
fix A B |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5341 |
assume \<section>: "A \<in> \<W>" "B \<in> \<W>" "A \<noteq> B" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5342 |
then have "disjnt A B" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5343 |
by (meson \<open>disjoint \<W>\<close> pairwiseD) |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5344 |
have "closedin (subtopology X (topspace X - C)) (\<Union>(\<V> - {T}))" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5345 |
using \<U>_def \<open>\<V> \<subseteq> \<U>\<close> closedin_connected_components_of \<open>finite \<V>\<close> |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5346 |
by (force simp add: intro!: closedin_Union) |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5347 |
with \<open>C \<subseteq> S\<close> have "openin (subtopology X (topspace X - S)) (topspace X - S - \<Union>(\<V> - {T}))" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5348 |
by (fastforce simp add: openin_closedin_eq closedin_subtopology Int_absorb1) |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5349 |
moreover have "\<And>V. V \<in> \<V> \<and> V\<noteq>T \<Longrightarrow> openin (subtopology X (topspace X - S)) (V - S)" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5350 |
using \<open>\<V> \<subseteq> \<U>\<close> ope\<U> |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5351 |
by (metis IntD2 Int_Diff inf.orderE openin_subset openin_subtopology) |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5352 |
ultimately have "openin (subtopology X (topspace X - S)) A" "openin (subtopology X (topspace X - S)) B" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5353 |
using \<section> \<W>_def by blast+ |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5354 |
with \<open>disjnt A B\<close> show "separatedin (subtopology X (topspace X - S)) A B" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5355 |
using separatedin_open_sets by blast |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5356 |
qed |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5357 |
then show ?thesis |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5358 |
by (simp add: pairwise_def separatedin_subtopology) |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5359 |
qed |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5360 |
ultimately show False |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5361 |
by (metis con eqpoll_iff_finite_card) |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5362 |
qed |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5363 |
obtain \<V> where "\<V> \<approx> {..<n} " "{} \<notin> \<V>" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5364 |
and pw\<V>: "pairwise (separatedin X) \<V>" and UV: "\<Union>\<V> = topspace X - (topspace X - \<Union>\<U>)" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5365 |
proof - |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5366 |
have "closedin X (topspace X - \<Union>\<U>)" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5367 |
using ope\<U> by blast |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5368 |
moreover have "C \<subseteq> topspace X - \<Union>\<U>" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5369 |
using \<open>C \<subseteq> topspace X\<close> connected_components_of_subset by (fastforce simp: \<U>_def) |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5370 |
moreover have "topspace X - \<Union>\<U> \<subseteq> S" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5371 |
using Union_connected_components_of [of "subtopology X (topspace X - C)"] \<open>C \<subseteq> S\<close> |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5372 |
by (auto simp: \<U>_def) |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5373 |
ultimately show thesis |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5374 |
by (metis C that) |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5375 |
qed |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5376 |
have "\<V> \<lesssim> \<U>" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5377 |
proof (rule lepoll_relational_full) |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5378 |
have "\<Union>\<V> = \<Union>\<U>" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5379 |
by (simp add: Sup_le_iff UV double_diff ope\<U> openin_subset) |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5380 |
then show "\<exists>U. U \<in> \<U> \<and> \<not> disjnt U V" if "V \<in> \<V>" for V |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5381 |
using that |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5382 |
by (metis \<open>{} \<notin> \<V>\<close> disjnt_Union1 disjnt_self_iff_empty) |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5383 |
show "C1 = C2" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5384 |
if "T \<in> \<U>" and "C1 \<in> \<V>" and "C2 \<in> \<V>" and "\<not> disjnt T C1" and "\<not> disjnt T C2" for T C1 C2 |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5385 |
proof (cases "C1=C2") |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5386 |
case False |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5387 |
then have "connectedin X T" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5388 |
using \<U>_def connectedin_connected_components_of connectedin_subtopology \<open>T \<in> \<U>\<close> by blast |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5389 |
have "T \<subseteq> C1 \<union> \<Union>(\<V> - {C1})" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5390 |
using \<open>\<Union>\<V> = \<Union>\<U>\<close> \<open>T \<in> \<U>\<close> by auto |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5391 |
with \<open>connectedin X T\<close> |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5392 |
have "\<not> separatedin X C1 (\<Union>(\<V> - {C1}))" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5393 |
unfolding connectedin_eq_not_separated_subset |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5394 |
by (smt (verit) that False disjnt_def UnionI disjnt_iff insertE insert_Diff) |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5395 |
with that show ?thesis |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5396 |
by (metis (no_types, lifting) \<open>\<V> \<approx> {..<n}\<close> eqpoll_iff_finite_card finite_Diff pairwiseD pairwise_alt pw\<V> separatedin_Union(1) separatedin_def) |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5397 |
qed auto |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5398 |
qed |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5399 |
then show False |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5400 |
by (metis \<open>\<V> \<approx> {..<n}\<close> card\<U> eqpoll_iff_finite_card leD lepoll_iff_card_le) |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5401 |
qed |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5402 |
qed |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5403 |
qed |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5404 |
|
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5405 |
lemma separation_by_closed_intermediates_eq_gen: |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5406 |
assumes "locally_connected_space X" "hereditarily normal_space X" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5407 |
shows "\<not> connectedin X (topspace X - S) \<longleftrightarrow> |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5408 |
(\<exists>C. closedin X C \<and> C \<subseteq> S \<and> |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5409 |
(\<forall>D. closedin X D \<and> C \<subseteq> D \<and> D \<subseteq> S \<longrightarrow> \<not> connectedin X (topspace X - D)))" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5410 |
(is "?lhs = ?rhs") |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5411 |
proof - |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5412 |
have *: "(\<exists>\<U>::'a set set. \<U> \<approx> {..<Suc (Suc 0)} \<and> P \<U>) \<longleftrightarrow> (\<exists>A B. A\<noteq>B \<and> P{A,B})" for P |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5413 |
by (metis One_nat_def eqpoll_doubleton_iff lessThan_Suc lessThan_empty_iff zero_neq_one) |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5414 |
have *: "(\<exists>C1 C2. separatedin X C1 C2 \<and> C1\<noteq>C2 \<and> C1\<noteq>{} \<and> C2\<noteq>{} \<and> C1 \<union> C2 = topspace X - S) \<longleftrightarrow> |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5415 |
(\<exists>C. closedin X C \<and> C \<subseteq> S \<and> |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5416 |
(\<forall>D. closedin X D \<and> C \<subseteq> D \<and> D \<subseteq> S |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5417 |
\<longrightarrow> (\<exists>C1 C2. separatedin X C1 C2 \<and> C1\<noteq>C2 \<and> C1\<noteq>{} \<and> C2\<noteq>{} \<and> C1 \<union> C2 = topspace X - D)))" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5418 |
using separation_by_closed_intermediates_eq_count [OF assms, of "Suc(Suc 0)" S] |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5419 |
apply (simp add: * pairwise_insert separatedin_sym cong: conj_cong) |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5420 |
apply (simp add: eq_sym_conv conj_ac) |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5421 |
done |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5422 |
with separatedin_refl |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5423 |
show ?thesis |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5424 |
apply (simp add: connectedin_eq_not_separated) |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5425 |
by (smt (verit, best) separatedin_refl) |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5426 |
qed |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5427 |
|
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5428 |
|
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5429 |
|
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5430 |
lemma lepoll_connnected_components_connectedin: |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5431 |
assumes "\<And>C. C \<in> \<U> \<Longrightarrow> connectedin X C" "\<Union>\<U> = topspace X" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5432 |
shows "connected_components_of X \<lesssim> \<U>" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5433 |
proof - |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5434 |
have "connected_components_of X \<lesssim> \<U> - {{}}" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5435 |
proof (rule lepoll_relational_full) |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5436 |
show "\<exists>U. U \<in> \<U> - {{}} \<and> U \<subseteq> V" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5437 |
if "V \<in> connected_components_of X" for V |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5438 |
using that unfolding connected_components_of_def image_iff |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5439 |
by (metis Union_iff assms connected_component_of_maximal empty_iff insert_Diff_single insert_iff) |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5440 |
show "V = V'" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5441 |
if "U \<in> \<U> - {{}}" "V \<in> connected_components_of X" "V' \<in> connected_components_of X" "U \<subseteq> V" "U \<subseteq> V'" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5442 |
for U V V' |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5443 |
by (metis DiffD2 disjointD insertCI le_inf_iff pairwise_disjoint_connected_components_of subset_empty that) |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5444 |
qed |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5445 |
also have "\<dots> \<lesssim> \<U>" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5446 |
by (simp add: subset_imp_lepoll) |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5447 |
finally show ?thesis . |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5448 |
qed |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5449 |
|
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5450 |
lemma lepoll_connected_components_alt: |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5451 |
"{..<n::nat} \<lesssim> connected_components_of X \<longleftrightarrow> |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5452 |
n = 0 \<or> (\<exists>\<U>. \<U> \<approx> {..<n} \<and> pairwise (separatedin X) \<U> \<and> {} \<notin> \<U> \<and> \<Union>\<U> = topspace X)" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5453 |
(is "?lhs \<longleftrightarrow> ?rhs") |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5454 |
proof (cases "n=0") |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5455 |
next |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5456 |
case False |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5457 |
show ?thesis |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5458 |
proof |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5459 |
assume L: ?lhs |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5460 |
with False show ?rhs |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5461 |
proof (induction n rule: less_induct) |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5462 |
case (less n) |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5463 |
show ?case |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5464 |
proof (cases "n\<le>1") |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5465 |
case True |
78336 | 5466 |
with less.prems have "topspace X \<noteq> {}" "n=1" |
5467 |
by (fastforce simp add: connected_components_of_def)+ |
|
5468 |
then have "{} \<notin> {topspace X}" |
|
5469 |
by blast |
|
5470 |
with \<open>n=1\<close> show ?thesis |
|
5471 |
by (simp add: eqpoll_iff_finite_card card_Suc_eq flip: ex_simps) |
|
78274
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5472 |
next |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5473 |
case False |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5474 |
then have "n-1 \<noteq> 0" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5475 |
by linarith |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5476 |
have n1_lesspoll: "{..<n-1} \<prec> {..<n}" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5477 |
using False lesspoll_iff_finite_card by fastforce |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5478 |
also have "\<dots> \<lesssim> connected_components_of X" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5479 |
using less by blast |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5480 |
finally have "{..<n-1} \<lesssim> connected_components_of X" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5481 |
using lesspoll_imp_lepoll by blast |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5482 |
then obtain \<U> where Ueq: "\<U> \<approx> {..<n-1}" and "{} \<notin> \<U>" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5483 |
and pwU: "pairwise (separatedin X) \<U>" and UU: "\<Union>\<U> = topspace X" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5484 |
by (meson \<open>n - 1 \<noteq> 0\<close> diff_less gr0I less zero_less_one) |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5485 |
show ?thesis |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5486 |
proof (cases "\<forall>C \<in> \<U>. connectedin X C") |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5487 |
case True |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5488 |
then show ?thesis |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5489 |
using lepoll_connnected_components_connectedin [of \<U> X] less.prems |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5490 |
by (metis UU Ueq lepoll_antisym lepoll_trans lepoll_trans2 lesspoll_def n1_lesspoll) |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5491 |
next |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5492 |
case False |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5493 |
with UU obtain C A B where ABC: "C \<in> \<U>" "A \<union> B = C" "A \<noteq> {}" "B \<noteq> {}" and sep: "separatedin X A B" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5494 |
by (fastforce simp add: connectedin_eq_not_separated) |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5495 |
define \<V> where "\<V> \<equiv> insert A (insert B (\<U> - {C}))" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5496 |
have "\<V> \<approx> {..<n}" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5497 |
proof - |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5498 |
have "A \<noteq> B" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5499 |
using \<open>B \<noteq> {}\<close> sep by auto |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5500 |
moreover obtain "A \<notin> \<U>" "B \<notin> \<U>" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5501 |
using pwU unfolding pairwise_def |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5502 |
by (metis ABC sep separatedin_Un(1) separatedin_refl separatedin_sym) |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5503 |
moreover have "card \<U> = n-1" "finite \<U>" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5504 |
using Ueq eqpoll_iff_finite_card by blast+ |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5505 |
ultimately |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5506 |
have "card (insert A (insert B (\<U> - {C}))) = n" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5507 |
using \<open>C \<in> \<U>\<close> by (auto simp add: card_insert_if) |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5508 |
then show ?thesis |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5509 |
using \<V>_def \<open>finite \<U>\<close> eqpoll_iff_finite_card by blast |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5510 |
qed |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5511 |
moreover have "{} \<notin> \<V>" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5512 |
using ABC \<V>_def \<open>{} \<notin> \<U>\<close> by blast |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5513 |
moreover have "\<Union>\<V> = topspace X" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5514 |
using ABC UU \<V>_def by auto |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5515 |
moreover have "pairwise (separatedin X) \<V>" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5516 |
using pwU sep ABC unfolding \<V>_def |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5517 |
apply (simp add: separatedin_sym pairwise_def) |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5518 |
by (metis member_remove remove_def separatedin_Un(1)) |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5519 |
ultimately show ?thesis |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5520 |
by blast |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5521 |
qed |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5522 |
qed |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5523 |
qed |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5524 |
next |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5525 |
assume ?rhs |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5526 |
then obtain \<U> where "\<U> \<approx> {..<n}" "{} \<notin> \<U>" and pwU: "pairwise (separatedin X) \<U>" and UU: "\<Union>\<U> = topspace X" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5527 |
using False by force |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5528 |
have "card (connected_components_of X) \<ge> n" if "finite (connected_components_of X)" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5529 |
proof - |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5530 |
have "\<U> \<lesssim> connected_components_of X" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5531 |
proof (rule lepoll_relational_full) |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5532 |
show "\<exists>T. T \<in> connected_components_of X \<and> \<not> disjnt T C" if "C \<in> \<U>" for C |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5533 |
by (metis that UU Union_connected_components_of Union_iff \<open>{} \<notin> \<U>\<close> disjnt_iff equals0I) |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5534 |
show "(C1::'a set) = C2" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5535 |
if "T \<in> connected_components_of X" and "C1 \<in> \<U>" "C2 \<in> \<U>" "\<not> disjnt T C1" "\<not> disjnt T C2" for T C1 C2 |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5536 |
proof (rule ccontr) |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5537 |
assume "C1 \<noteq> C2" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5538 |
then have "connectedin X T" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5539 |
by (simp add: connectedin_connected_components_of that(1)) |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5540 |
moreover have "\<not> separatedin X C1 (\<Union>(\<U> - {C1}))" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5541 |
using \<open>connectedin X T\<close> pwU unfolding pairwise_def |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5542 |
by (smt (verit) Sup_upper UU Union_connected_components_of \<open>C1 \<noteq> C2\<close> complete_lattice_class.Sup_insert connectedin_subset_separated_union disjnt_subset2 disjnt_sym insert_Diff separatedin_imp_disjoint that) |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5543 |
ultimately show False |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5544 |
using \<open>\<U> \<approx> {..<n}\<close> |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5545 |
apply (simp add: connectedin_eq_not_separated_subset eqpoll_iff_finite_card) |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5546 |
by (metis Sup_upper UU finite_Diff pairwise_alt pwU separatedin_Union(1) that(2)) |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5547 |
qed |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5548 |
qed |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5549 |
then show ?thesis |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5550 |
by (metis \<open>\<U> \<approx> {..<n}\<close> eqpoll_iff_finite_card lepoll_iff_card_le that) |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5551 |
qed |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5552 |
then show ?lhs |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5553 |
by (metis card_lessThan finite_lepoll_infinite finite_lessThan lepoll_iff_card_le) |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5554 |
qed |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5555 |
qed auto |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5556 |
|
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5557 |
|
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5558 |
subsection\<open>A perfect set in common cases must have at least the cardinality of the continuum\<close> |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5559 |
|
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5560 |
lemma (in Metric_space) lepoll_perfect_set: |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5561 |
assumes "mcomplete" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5562 |
and "mtopology derived_set_of S = S" "S \<noteq> {}" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5563 |
shows "(UNIV::real set) \<lesssim> S" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5564 |
proof - |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5565 |
have "S \<subseteq> M" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5566 |
using assms(2) derived_set_of_infinite_mball by blast |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5567 |
have "(UNIV::real set) \<lesssim> (UNIV::nat set set)" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5568 |
using eqpoll_imp_lepoll eqpoll_sym nat_sets_eqpoll_reals by blast |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5569 |
also have "\<dots> \<lesssim> S" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5570 |
proof - |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5571 |
have "\<exists>y z \<delta>. y \<in> S \<and> z \<in> S \<and> 0 < \<delta> \<and> \<delta> < \<epsilon>/2 \<and> |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5572 |
mcball y \<delta> \<subseteq> mcball x \<epsilon> \<and> mcball z \<delta> \<subseteq> mcball x \<epsilon> \<and> disjnt (mcball y \<delta>) (mcball z \<delta>)" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5573 |
if "x \<in> S" "0 < \<epsilon>" for x \<epsilon> |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5574 |
proof - |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5575 |
define S' where "S' \<equiv> S \<inter> mball x (\<epsilon>/4)" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5576 |
have "infinite S'" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5577 |
using derived_set_of_infinite_mball [of S] assms that S'_def |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5578 |
by (smt (verit, ccfv_SIG) mem_Collect_eq zero_less_divide_iff) |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5579 |
then have "\<And>x y z. \<not> (S' \<subseteq> {x,y,z})" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5580 |
using finite_subset by auto |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5581 |
then obtain l r where lr: "l \<in> S'" "r \<in> S'" "l\<noteq>r" "l\<noteq>x" "r\<noteq>x" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5582 |
by (metis insert_iff subsetI) |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5583 |
show ?thesis |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5584 |
proof (intro exI conjI) |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5585 |
show "l \<in> S" "r \<in> S" "d l r / 3 > 0" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5586 |
using lr by (auto simp: S'_def) |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5587 |
show "d l r / 3 < \<epsilon>/2" "mcball l (d l r / 3) \<subseteq> mcball x \<epsilon>" "mcball r (d l r / 3) \<subseteq> mcball x \<epsilon>" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5588 |
using lr by (clarsimp simp: S'_def, smt (verit) commute triangle'')+ |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5589 |
show "disjnt (mcball l (d l r / 3)) (mcball r (d l r / 3))" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5590 |
using lr by (simp add: S'_def disjnt_iff) (smt (verit, best) mdist_pos_less triangle') |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5591 |
qed |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5592 |
qed |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5593 |
then obtain l r \<delta> |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5594 |
where lrS: "\<And>x \<epsilon>. \<lbrakk>x \<in> S; 0 < \<epsilon>\<rbrakk> \<Longrightarrow> l x \<epsilon> \<in> S \<and> r x \<epsilon> \<in> S" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5595 |
and \<delta>: "\<And>x \<epsilon>. \<lbrakk>x \<in> S; 0 < \<epsilon>\<rbrakk> \<Longrightarrow> 0 < \<delta> x \<epsilon> \<and> \<delta> x \<epsilon> < \<epsilon>/2" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5596 |
and "\<And>x \<epsilon>. \<lbrakk>x \<in> S; 0 < \<epsilon>\<rbrakk> \<Longrightarrow> mcball (l x \<epsilon>) (\<delta> x \<epsilon>) \<subseteq> mcball x \<epsilon> \<and> mcball (r x \<epsilon>) (\<delta> x \<epsilon>) \<subseteq> mcball x \<epsilon> \<and> |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5597 |
disjnt (mcball (l x \<epsilon>) (\<delta> x \<epsilon>)) (mcball (r x \<epsilon>) (\<delta> x \<epsilon>))" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5598 |
by metis |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5599 |
then have lr_mcball: "\<And>x \<epsilon>. \<lbrakk>x \<in> S; 0 < \<epsilon>\<rbrakk> \<Longrightarrow> mcball (l x \<epsilon>) (\<delta> x \<epsilon>) \<subseteq> mcball x \<epsilon> \<and> mcball (r x \<epsilon>) (\<delta> x \<epsilon>) \<subseteq> mcball x \<epsilon> " |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5600 |
and lr_disjnt: "\<And>x \<epsilon>. \<lbrakk>x \<in> S; 0 < \<epsilon>\<rbrakk> \<Longrightarrow> disjnt (mcball (l x \<epsilon>) (\<delta> x \<epsilon>)) (mcball (r x \<epsilon>) (\<delta> x \<epsilon>))" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5601 |
by metis+ |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5602 |
obtain a where "a \<in> S" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5603 |
using \<open>S \<noteq> {}\<close> by blast |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5604 |
define xe where "xe \<equiv> |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5605 |
\<lambda>B. rec_nat (a,1) (\<lambda>n (x,\<gamma>). ((if n\<in>B then r else l) x \<gamma>, \<delta> x \<gamma>))" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5606 |
have [simp]: "xe b 0 = (a,1)" for b |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5607 |
by (simp add: xe_def) |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5608 |
have "xe B (Suc n) = (let (x,\<gamma>) = xe B n in ((if n\<in>B then r else l) x \<gamma>, \<delta> x \<gamma>))" for B n |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5609 |
by (simp add: xe_def) |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5610 |
define x where "x \<equiv> \<lambda>B n. fst (xe B n)" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5611 |
define \<gamma> where "\<gamma> \<equiv> \<lambda>B n. snd (xe B n)" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5612 |
have [simp]: "x B 0 = a" "\<gamma> B 0 = 1" for B |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5613 |
by (simp_all add: x_def \<gamma>_def xe_def) |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5614 |
have x_Suc[simp]: "x B (Suc n) = ((if n\<in>B then r else l) (x B n) (\<gamma> B n))" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5615 |
and \<gamma>_Suc[simp]: "\<gamma> B (Suc n) = \<delta> (x B n) (\<gamma> B n)" for B n |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5616 |
by (simp_all add: x_def \<gamma>_def xe_def split: prod.split) |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5617 |
interpret Submetric M d S |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5618 |
proof qed (use \<open>S \<subseteq> M\<close> in metis) |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5619 |
have "closedin mtopology S" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5620 |
by (metis assms(2) closure_of closure_of_eq inf.absorb_iff2 subset subset_Un_eq subset_refl topspace_mtopology) |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5621 |
with \<open>mcomplete\<close> |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5622 |
have "sub.mcomplete" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5623 |
by (metis closedin_mcomplete_imp_mcomplete) |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5624 |
have *: "x B n \<in> S \<and> \<gamma> B n > 0" for B n |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5625 |
by (induction n) (auto simp: \<open>a \<in> S\<close> lrS \<delta>) |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5626 |
with subset have E: "x B n \<in> M" for B n |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5627 |
by blast |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5628 |
have \<gamma>_le: "\<gamma> B n \<le> (1/2)^n" for B n |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5629 |
proof(induction n) |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5630 |
case 0 then show ?case by auto |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5631 |
next |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5632 |
case (Suc n) |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5633 |
then show ?case |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5634 |
by simp (smt (verit) "*" \<delta> field_sum_of_halves) |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5635 |
qed |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5636 |
{ fix B |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5637 |
have "\<And>n. sub.mcball (x B (Suc n)) (\<gamma> B (Suc n)) \<subseteq> sub.mcball (x B n) (\<gamma> B n)" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5638 |
by (smt (verit, best) "*" Int_iff \<gamma>_Suc x_Suc in_mono lr_mcball mcball_submetric_eq subsetI) |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5639 |
then have mon: "monotone (\<le>) (\<lambda>x y. y \<subseteq> x) (\<lambda>n. sub.mcball (x B n) (\<gamma> B n))" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5640 |
by (simp add: decseq_SucI) |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5641 |
have "\<exists>n a. sub.mcball (x B n) (\<gamma> B n) \<subseteq> sub.mcball a \<epsilon>" if "\<epsilon>>0" for \<epsilon> |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5642 |
proof - |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5643 |
obtain n where "(1/2)^n < \<epsilon>" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5644 |
using \<open>0 < \<epsilon>\<close> real_arch_pow_inv by force |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5645 |
with \<gamma>_le have \<epsilon>: "\<gamma> B n \<le> \<epsilon>" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5646 |
by (smt (verit)) |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5647 |
show ?thesis |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5648 |
proof (intro exI) |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5649 |
show "sub.mcball (x B n) (\<gamma> B n) \<subseteq> sub.mcball (x B n) \<epsilon>" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5650 |
by (simp add: \<epsilon> sub.mcball_subset_concentric) |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5651 |
qed |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5652 |
qed |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5653 |
then have "\<exists>l. l \<in> S \<and> (\<Inter>n. sub.mcball (x B n) (\<gamma> B n)) = {l}" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5654 |
using \<open>sub.mcomplete\<close> mon |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5655 |
unfolding sub.mcomplete_nest_sing |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5656 |
apply (drule_tac x="\<lambda>n. sub.mcball (x B n) (\<gamma> B n)" in spec) |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5657 |
by (meson * order.asym sub.closedin_mcball sub.mcball_eq_empty) |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5658 |
} |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5659 |
then obtain z where z: "\<And>B. z B \<in> S \<and> (\<Inter>n. sub.mcball (x B n) (\<gamma> B n)) = {z B}" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5660 |
by metis |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5661 |
show ?thesis |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5662 |
unfolding lepoll_def |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5663 |
proof (intro exI conjI) |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5664 |
show "inj z" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5665 |
proof (rule inj_onCI) |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5666 |
fix B C |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5667 |
assume eq: "z B = z C" and "B \<noteq> C" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5668 |
then have ne: "sym_diff B C \<noteq> {}" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5669 |
by blast |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5670 |
define n where "n \<equiv> LEAST k. k \<in> (sym_diff B C)" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5671 |
with ne have n: "n \<in> sym_diff B C" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5672 |
by (metis Inf_nat_def1 LeastI) |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5673 |
then have non: "n \<in> B \<longleftrightarrow> n \<notin> C" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5674 |
by blast |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5675 |
have H: "z C \<in> sub.mcball (x B (Suc n)) (\<gamma> B (Suc n)) \<and> z C \<in> sub.mcball (x C (Suc n)) (\<gamma> C (Suc n))" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5676 |
using z [of B] z [of C] apply (simp add: lrS set_eq_iff non *) |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5677 |
by (smt (verit, best) \<gamma>_Suc eq non x_Suc) |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5678 |
have "k \<in> B \<longleftrightarrow> k \<in> C" if "k<n" for k |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5679 |
using that unfolding n_def by (meson DiffI UnCI not_less_Least) |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5680 |
moreover have "(\<forall>m. m < p \<longrightarrow> (m \<in> B \<longleftrightarrow> m \<in> C)) \<Longrightarrow> x B p = x C p \<and> \<gamma> B p = \<gamma> C p" for p |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5681 |
by (induction p) auto |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5682 |
ultimately have "x B n = x C n" "\<gamma> B n = \<gamma> C n" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5683 |
by blast+ |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5684 |
then show False |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5685 |
using lr_disjnt * H non |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5686 |
by (smt (verit) IntD2 \<gamma>_Suc disjnt_iff mcball_submetric_eq x_Suc) |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5687 |
qed |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5688 |
show "range z \<subseteq> S" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5689 |
using z by blast |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5690 |
qed |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5691 |
qed |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5692 |
finally show ?thesis . |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5693 |
qed |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5694 |
|
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5695 |
lemma lepoll_perfect_set_aux: |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5696 |
assumes lcX: "locally_compact_space X" and hsX: "Hausdorff_space X" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5697 |
and eq: "X derived_set_of topspace X = topspace X" and "topspace X \<noteq> {}" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5698 |
shows "(UNIV::real set) \<lesssim> topspace X" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5699 |
proof - |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5700 |
have "(UNIV::real set) \<lesssim> (UNIV::nat set set)" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5701 |
using eqpoll_imp_lepoll eqpoll_sym nat_sets_eqpoll_reals by blast |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5702 |
also have "\<dots> \<lesssim> topspace X" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5703 |
proof - |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5704 |
obtain z where z: "z \<in> topspace X" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5705 |
using assms by blast |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5706 |
then obtain U K where "openin X U" "compactin X K" "U \<noteq> {}" "U \<subseteq> K" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5707 |
by (metis emptyE lcX locally_compact_space_def) |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5708 |
then have "closedin X K" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5709 |
by (simp add: compactin_imp_closedin hsX) |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5710 |
have intK_ne: "X interior_of K \<noteq> {}" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5711 |
using \<open>U \<noteq> {}\<close> \<open>U \<subseteq> K\<close> \<open>openin X U\<close> interior_of_eq_empty by blast |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5712 |
have "\<exists>D E. closedin X D \<and> D \<subseteq> K \<and> X interior_of D \<noteq> {} \<and> |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5713 |
closedin X E \<and> E \<subseteq> K \<and> X interior_of E \<noteq> {} \<and> |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5714 |
disjnt D E \<and> D \<subseteq> C \<and> E \<subseteq> C" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5715 |
if "closedin X C" "C \<subseteq> K" and C: "X interior_of C \<noteq> {}" for C |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5716 |
proof - |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5717 |
obtain z where z: "z \<in> X interior_of C" "z \<in> topspace X" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5718 |
using C interior_of_subset_topspace by fastforce |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5719 |
obtain x y where "x \<in> X interior_of C" "y \<in> X interior_of C" "x\<noteq>y" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5720 |
by (metis z eq in_derived_set_of openin_interior_of) |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5721 |
then have "x \<in> topspace X" "y \<in> topspace X" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5722 |
using interior_of_subset_topspace by force+ |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5723 |
with hsX obtain V W where "openin X V" "openin X W" "x \<in> V" "y \<in> W" "disjnt V W" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5724 |
by (metis Hausdorff_space_def \<open>x \<noteq> y\<close>) |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5725 |
have *: "\<And>W x. openin X W \<and> x \<in> W |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5726 |
\<Longrightarrow> \<exists>U V. openin X U \<and> closedin X V \<and> x \<in> U \<and> U \<subseteq> V \<and> V \<subseteq> W" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5727 |
using lcX hsX locally_compact_Hausdorff_imp_regular_space neighbourhood_base_of_closedin neighbourhood_base_of |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5728 |
by metis |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5729 |
obtain M D where MD: "openin X M" "closedin X D" "y \<in> M" "M \<subseteq> D" "D \<subseteq> X interior_of C \<inter> W" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5730 |
using * [of "X interior_of C \<inter> W" y] |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5731 |
using \<open>openin X W\<close> \<open>y \<in> W\<close> \<open>y \<in> X interior_of C\<close> by fastforce |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5732 |
obtain N E where NE: "openin X N" "closedin X E" "x \<in> N" "N \<subseteq> E" "E \<subseteq> X interior_of C \<inter> V" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5733 |
using * [of "X interior_of C \<inter> V" x] |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5734 |
using \<open>openin X V\<close> \<open>x \<in> V\<close> \<open>x \<in> X interior_of C\<close> by fastforce |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5735 |
show ?thesis |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5736 |
proof (intro exI conjI) |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5737 |
show "X interior_of D \<noteq> {}" "X interior_of E \<noteq> {}" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5738 |
using MD NE by (fastforce simp: interior_of_def)+ |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5739 |
show "disjnt D E" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5740 |
by (meson MD(5) NE(5) \<open>disjnt V W\<close> disjnt_subset1 disjnt_sym le_inf_iff) |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5741 |
qed (use MD NE \<open>C \<subseteq> K\<close> interior_of_subset in force)+ |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5742 |
qed |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5743 |
then obtain L R where |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5744 |
LR: "\<And>C. \<lbrakk>closedin X C; C \<subseteq> K; X interior_of C \<noteq> {}\<rbrakk> |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5745 |
\<Longrightarrow> closedin X (L C) \<and> (L C) \<subseteq> K \<and> X interior_of (L C) \<noteq> {} \<and> |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5746 |
closedin X (R C) \<and> (R C) \<subseteq> K \<and> X interior_of (R C) \<noteq> {}" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5747 |
and disjLR: "\<And>C. \<lbrakk>closedin X C; C \<subseteq> K; X interior_of C \<noteq> {}\<rbrakk> |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5748 |
\<Longrightarrow> disjnt (L C) (R C) \<and> (L C) \<subseteq> C \<and> (R C) \<subseteq> C" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5749 |
by metis |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5750 |
define d where "d \<equiv> \<lambda>B. rec_nat K (\<lambda>n. if n \<in> B then R else L)" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5751 |
have d0[simp]: "d B 0 = K" for B |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5752 |
by (simp add: d_def) |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5753 |
have [simp]: "d B (Suc n) = (if n \<in> B then R else L) (d B n)" for B n |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5754 |
by (simp add: d_def) |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5755 |
have d_correct: "closedin X (d B n) \<and> d B n \<subseteq> K \<and> X interior_of (d B n) \<noteq> {}" for B n |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5756 |
proof (induction n) |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5757 |
case 0 |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5758 |
then show ?case by (auto simp: \<open>closedin X K\<close> intK_ne) |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5759 |
next |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5760 |
case (Suc n) with LR show ?case by auto |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5761 |
qed |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5762 |
have "(\<Inter>n. d B n) \<noteq> {}" for B |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5763 |
proof (rule compact_space_imp_nest) |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5764 |
show "compact_space (subtopology X K)" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5765 |
by (simp add: \<open>compactin X K\<close> compact_space_subtopology) |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5766 |
show "closedin (subtopology X K) (d B n)" for n :: nat |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5767 |
by (simp add: closedin_subset_topspace d_correct) |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5768 |
show "d B n \<noteq> {}" for n :: nat |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5769 |
by (metis d_correct interior_of_empty) |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5770 |
show "antimono (d B)" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5771 |
proof (rule antimonoI [OF transitive_stepwise_le]) |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5772 |
fix n |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5773 |
show "d B (Suc n) \<subseteq> d B n" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5774 |
by (simp add: d_correct disjLR) |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5775 |
qed auto |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5776 |
qed |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5777 |
then obtain x where x: "\<And>B. x B \<in> (\<Inter>n. d B n)" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5778 |
unfolding set_eq_iff by (metis empty_iff) |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5779 |
show ?thesis |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5780 |
unfolding lepoll_def |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5781 |
proof (intro exI conjI) |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5782 |
show "inj x" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5783 |
proof (rule inj_onCI) |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5784 |
fix B C |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5785 |
assume eq: "x B = x C" and "B\<noteq>C" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5786 |
then have ne: "sym_diff B C \<noteq> {}" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5787 |
by blast |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5788 |
define n where "n \<equiv> LEAST k. k \<in> (sym_diff B C)" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5789 |
with ne have n: "n \<in> sym_diff B C" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5790 |
by (metis Inf_nat_def1 LeastI) |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5791 |
then have non: "n \<in> B \<longleftrightarrow> n \<notin> C" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5792 |
by blast |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5793 |
have "k \<in> B \<longleftrightarrow> k \<in> C" if "k<n" for k |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5794 |
using that unfolding n_def by (meson DiffI UnCI not_less_Least) |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5795 |
moreover have "(\<forall>m. m < p \<longrightarrow> (m \<in> B \<longleftrightarrow> m \<in> C)) \<Longrightarrow> d B p = d C p" for p |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5796 |
by (induction p) auto |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5797 |
ultimately have "d B n = d C n" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5798 |
by blast |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5799 |
then have "disjnt (d B (Suc n)) (d C (Suc n))" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5800 |
by (simp add: d_correct disjLR disjnt_sym non) |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5801 |
then show False |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5802 |
by (metis InterE disjnt_iff eq rangeI x) |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5803 |
qed |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5804 |
show "range x \<subseteq> topspace X" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5805 |
using x d0 \<open>compactin X K\<close> compactin_subset_topspace d_correct by fastforce |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5806 |
qed |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5807 |
qed |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5808 |
finally show ?thesis . |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5809 |
qed |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5810 |
|
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5811 |
lemma lepoll_perfect_set: |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5812 |
assumes X: "completely_metrizable_space X \<or> locally_compact_space X \<and> Hausdorff_space X" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5813 |
and S: "X derived_set_of S = S" "S \<noteq> {}" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5814 |
shows "(UNIV::real set) \<lesssim> S" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5815 |
using X |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5816 |
proof |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5817 |
assume "completely_metrizable_space X" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5818 |
with assms show "(UNIV::real set) \<lesssim> S" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5819 |
by (metis Metric_space.lepoll_perfect_set completely_metrizable_space_def) |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5820 |
next |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5821 |
assume "locally_compact_space X \<and> Hausdorff_space X" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5822 |
then show "(UNIV::real set) \<lesssim> S" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5823 |
using lepoll_perfect_set_aux [of "subtopology X S"] |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5824 |
by (metis Hausdorff_space_subtopology S closedin_derived_set_of closedin_subset derived_set_of_subtopology |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5825 |
locally_compact_space_closed_subset subtopology_topspace topspace_subtopology topspace_subtopology_subset) |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5826 |
qed |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5827 |
|
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5828 |
|
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5829 |
|
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5830 |
|
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5831 |
lemma Kuratowski_aux1: |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5832 |
assumes "\<And>S T. R S T \<Longrightarrow> R T S" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5833 |
shows "(\<forall>S T n. R S T \<longrightarrow> (f S \<approx> {..<n::nat} \<longleftrightarrow> f T \<approx> {..<n::nat})) \<longleftrightarrow> |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5834 |
(\<forall>n S T. R S T \<longrightarrow> {..<n::nat} \<lesssim> f S \<longrightarrow> {..<n::nat} \<lesssim> f T)" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5835 |
(is "?lhs = ?rhs") |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5836 |
proof |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5837 |
assume ?lhs then show ?rhs |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5838 |
by (meson eqpoll_iff_finite_card eqpoll_sym finite_lepoll_infinite finite_lessThan lepoll_trans2) |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5839 |
next |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5840 |
assume ?rhs then show ?lhs |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5841 |
by (smt (verit, best) lepoll_iff_finite_card assms eqpoll_iff_finite_card finite_lepoll_infinite |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5842 |
finite_lessThan le_Suc_eq lepoll_antisym lepoll_iff_card_le not_less_eq_eq) |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5843 |
qed |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5844 |
|
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5845 |
lemma Kuratowski_aux2: |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5846 |
"pairwise (separatedin (subtopology X (topspace X - S))) \<U> \<and> {} \<notin> \<U> \<and> |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5847 |
\<Union>\<U> = topspace(subtopology X (topspace X - S)) \<longleftrightarrow> |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5848 |
pairwise (separatedin X) \<U> \<and> {} \<notin> \<U> \<and> \<Union>\<U> = topspace X - S" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5849 |
by (auto simp: pairwise_def separatedin_subtopology) |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5850 |
|
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5851 |
proposition Kuratowski_component_number_invariance_aux: |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5852 |
assumes "compact_space X" and HsX: "Hausdorff_space X" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5853 |
and lcX: "locally_connected_space X" and hnX: "hereditarily normal_space X" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5854 |
and hom: "(subtopology X S) homeomorphic_space (subtopology X T)" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5855 |
and leXS: "{..<n::nat} \<lesssim> connected_components_of (subtopology X (topspace X - S))" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5856 |
assumes \<section>: "\<And>S T. |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5857 |
\<lbrakk>closedin X S; closedin X T; (subtopology X S) homeomorphic_space (subtopology X T); |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5858 |
{..<n::nat} \<lesssim> connected_components_of (subtopology X (topspace X - S))\<rbrakk> |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5859 |
\<Longrightarrow> {..<n::nat} \<lesssim> connected_components_of (subtopology X (topspace X - T))" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5860 |
shows "{..<n::nat} \<lesssim> connected_components_of (subtopology X (topspace X - T))" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5861 |
proof (cases "n=0") |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5862 |
case False |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5863 |
obtain f g where homf: "homeomorphic_map (subtopology X S) (subtopology X T) f" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5864 |
and homg: "homeomorphic_map (subtopology X T) (subtopology X S) g" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5865 |
and gf: "\<And>x. x \<in> topspace (subtopology X S) \<Longrightarrow> g(f x) = x" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5866 |
and fg: "\<And>y. y \<in> topspace (subtopology X T) \<Longrightarrow> f(g y) = y" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5867 |
and f: "f \<in> topspace (subtopology X S) \<rightarrow> topspace (subtopology X T)" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5868 |
and g: "g \<in> topspace (subtopology X T) \<rightarrow> topspace (subtopology X S)" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5869 |
using homeomorphic_space_unfold hom by metis |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5870 |
obtain C where "closedin X C" "C \<subseteq> S" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5871 |
and C: "\<And>D. \<lbrakk>closedin X D; C \<subseteq> D; D \<subseteq> S\<rbrakk> |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5872 |
\<Longrightarrow> \<exists>\<U>. \<U> \<approx> {..<n} \<and> pairwise (separatedin X) \<U> \<and> {} \<notin> \<U> \<and> \<Union>\<U> = topspace X - D" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5873 |
using separation_by_closed_intermediates_eq_count [of X n S] assms |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5874 |
by (smt (verit, ccfv_threshold) False Kuratowski_aux2 lepoll_connected_components_alt) |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5875 |
have "\<exists>C. closedin X C \<and> C \<subseteq> T \<and> |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5876 |
(\<forall>D. closedin X D \<and> C \<subseteq> D \<and> D \<subseteq> T |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5877 |
\<longrightarrow> (\<exists>\<U>. \<U> \<approx> {..<n} \<and> pairwise (separatedin X) \<U> \<and> |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5878 |
{} \<notin> \<U> \<and> \<Union>\<U> = topspace X - D))" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5879 |
proof (intro exI, intro conjI strip) |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5880 |
have "compactin X (f ` C)" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5881 |
by (meson \<open>C \<subseteq> S\<close> \<open>closedin X C\<close> assms(1) closedin_compact_space compactin_subtopology homeomorphic_map_compactness_eq homf) |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5882 |
then show "closedin X (f ` C)" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5883 |
using \<open>Hausdorff_space X\<close> compactin_imp_closedin by blast |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5884 |
show "f ` C \<subseteq> T" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5885 |
by (meson \<open>C \<subseteq> S\<close> \<open>closedin X C\<close> closedin_imp_subset closedin_subset_topspace homeomorphic_map_closedness_eq homf) |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5886 |
fix D' |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5887 |
assume D': "closedin X D' \<and> f ` C \<subseteq> D' \<and> D' \<subseteq> T" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5888 |
define D where "D \<equiv> g ` D'" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5889 |
have "compactin X D" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5890 |
unfolding D_def |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5891 |
by (meson D' \<open>compact_space X\<close> closedin_compact_space compactin_subtopology homeomorphic_map_compactness_eq homg) |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5892 |
then have "closedin X D" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5893 |
by (simp add: assms(2) compactin_imp_closedin) |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5894 |
moreover have "C \<subseteq> D" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5895 |
using D' D_def \<open>C \<subseteq> S\<close> \<open>closedin X C\<close> closedin_subset gf image_iff by fastforce |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5896 |
moreover have "D \<subseteq> S" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5897 |
by (metis D' D_def assms(1) closedin_compact_space compactin_subtopology homeomorphic_map_compactness_eq homg) |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5898 |
ultimately obtain \<U> where "\<U> \<approx> {..<n}" "pairwise (separatedin X) \<U>" "{} \<notin> \<U>" "\<Union>\<U> = topspace X - D" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5899 |
using C by meson |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5900 |
moreover have "(subtopology X D) homeomorphic_space (subtopology X D')" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5901 |
unfolding homeomorphic_space_def |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5902 |
proof (intro exI) |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5903 |
have "subtopology X D = subtopology (subtopology X S) D" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5904 |
by (simp add: \<open>D \<subseteq> S\<close> inf.absorb2 subtopology_subtopology) |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5905 |
moreover have "subtopology X D' = subtopology (subtopology X T) D'" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5906 |
by (simp add: D' inf.absorb2 subtopology_subtopology) |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5907 |
moreover have "homeomorphic_maps (subtopology X T) (subtopology X S) g f" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5908 |
by (simp add: fg gf homeomorphic_maps_map homf homg) |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5909 |
ultimately |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5910 |
have "homeomorphic_maps (subtopology X D') (subtopology X D) g f" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5911 |
by (metis D' D_def \<open>closedin X D\<close> closedin_subset homeomorphic_maps_subtopologies topspace_subtopology Int_absorb1) |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5912 |
then show "homeomorphic_maps (subtopology X D) (subtopology X D') f g" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5913 |
using homeomorphic_maps_sym by blast |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5914 |
qed |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5915 |
ultimately show "\<exists>\<U>. \<U> \<approx> {..<n} \<and> pairwise (separatedin X) \<U> \<and> {} \<notin> \<U> \<and> \<Union> \<U> = topspace X - D'" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5916 |
by (smt (verit, ccfv_SIG) \<section> D' False \<open>closedin X D\<close> Kuratowski_aux2 lepoll_connected_components_alt) |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5917 |
qed |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5918 |
then have "\<exists>\<U>. \<U> \<approx> {..<n} \<and> |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5919 |
pairwise (separatedin (subtopology X (topspace X - T))) \<U> \<and> {} \<notin> \<U> \<and> \<Union>\<U> = topspace X - T" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5920 |
using separation_by_closed_intermediates_eq_count [of X n T] Kuratowski_aux2 lcX hnX by auto |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5921 |
with False show ?thesis |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5922 |
using lepoll_connected_components_alt by fastforce |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5923 |
qed auto |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5924 |
|
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5925 |
|
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5926 |
theorem Kuratowski_component_number_invariance: |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5927 |
assumes "compact_space X" "Hausdorff_space X" "locally_connected_space X" "hereditarily normal_space X" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5928 |
shows "((\<forall>S T n. |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5929 |
closedin X S \<and> closedin X T \<and> |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5930 |
(subtopology X S) homeomorphic_space (subtopology X T) |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5931 |
\<longrightarrow> (connected_components_of |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5932 |
(subtopology X (topspace X - S)) \<approx> {..<n::nat} \<longleftrightarrow> |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5933 |
connected_components_of |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5934 |
(subtopology X (topspace X - T)) \<approx> {..<n::nat})) \<longleftrightarrow> |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5935 |
(\<forall>S T n. |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5936 |
(subtopology X S) homeomorphic_space (subtopology X T) |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5937 |
\<longrightarrow> (connected_components_of |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5938 |
(subtopology X (topspace X - S)) \<approx> {..<n::nat} \<longleftrightarrow> |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5939 |
connected_components_of |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5940 |
(subtopology X (topspace X - T)) \<approx> {..<n::nat})))" |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5941 |
(is "?lhs = ?rhs") |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5942 |
proof |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5943 |
assume L: ?lhs |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5944 |
then show ?rhs |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5945 |
apply (subst (asm) Kuratowski_aux1, use homeomorphic_space_sym in blast) |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5946 |
apply (subst Kuratowski_aux1, use homeomorphic_space_sym in blast) |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5947 |
apply (blast intro: Kuratowski_component_number_invariance_aux assms) |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5948 |
done |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5949 |
qed blast |
f44aec9a6894
Last of the HOL Light metric space imports, and some supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
78256
diff
changeset
|
5950 |
|
78202 | 5951 |
end |