(* Author: L C Paulson, University of Cambridge
Author: Amine Chaieb, University of Cambridge
Author: Robert Himmelmann, TU Muenchen
Author: Brian Huffman, Portland State University
*)
section \<open>Abstract Topology 2\<close>
theory Abstract_Topology_2
imports
Elementary_Topology
Abstract_Topology
"HOL-Library.Indicator_Function"
begin
text \<open>Combination of Elementary and Abstract Topology\<close>
(* FIXME: move elsewhere *)
lemma approachable_lt_le: "(\<exists>(d::real) > 0. \<forall>x. f x < d \<longrightarrow> P x) \<longleftrightarrow> (\<exists>d>0. \<forall>x. f x \<le> d \<longrightarrow> P x)"
apply auto
apply (rule_tac x="d/2" in exI)
apply auto
done
lemma approachable_lt_le2: \<comment> \<open>like the above, but pushes aside an extra formula\<close>
"(\<exists>(d::real) > 0. \<forall>x. Q x \<longrightarrow> f x < d \<longrightarrow> P x) \<longleftrightarrow> (\<exists>d>0. \<forall>x. f x \<le> d \<longrightarrow> Q x \<longrightarrow> P x)"
apply auto
apply (rule_tac x="d/2" in exI, auto)
done
lemma triangle_lemma:
fixes x y z :: real
assumes x: "0 \<le> x"
and y: "0 \<le> y"
and z: "0 \<le> z"
and xy: "x\<^sup>2 \<le> y\<^sup>2 + z\<^sup>2"
shows "x \<le> y + z"
proof -
have "y\<^sup>2 + z\<^sup>2 \<le> y\<^sup>2 + 2 * y * z + z\<^sup>2"
using z y by simp
with xy have th: "x\<^sup>2 \<le> (y + z)\<^sup>2"
by (simp add: power2_eq_square field_simps)
from y z have yz: "y + z \<ge> 0"
by arith
from power2_le_imp_le[OF th yz] show ?thesis .
qed
lemma isCont_indicator:
fixes x :: "'a::t2_space"
shows "isCont (indicator A :: 'a \<Rightarrow> real) x = (x \<notin> frontier A)"
proof auto
fix x
assume cts_at: "isCont (indicator A :: 'a \<Rightarrow> real) x" and fr: "x \<in> frontier A"
with continuous_at_open have 1: "\<forall>V::real set. open V \<and> indicator A x \<in> V \<longrightarrow>
(\<exists>U::'a set. open U \<and> x \<in> U \<and> (\<forall>y\<in>U. indicator A y \<in> V))" by auto
show False
proof (cases "x \<in> A")
assume x: "x \<in> A"
hence "indicator A x \<in> ({0<..<2} :: real set)" by simp
hence "\<exists>U. open U \<and> x \<in> U \<and> (\<forall>y\<in>U. indicator A y \<in> ({0<..<2} :: real set))"
using 1 open_greaterThanLessThan by blast
then guess U .. note U = this
hence "\<forall>y\<in>U. indicator A y > (0::real)"
unfolding greaterThanLessThan_def by auto
hence "U \<subseteq> A" using indicator_eq_0_iff by force
hence "x \<in> interior A" using U interiorI by auto
thus ?thesis using fr unfolding frontier_def by simp
next
assume x: "x \<notin> A"
hence "indicator A x \<in> ({-1<..<1} :: real set)" by simp
hence "\<exists>U. open U \<and> x \<in> U \<and> (\<forall>y\<in>U. indicator A y \<in> ({-1<..<1} :: real set))"
using 1 open_greaterThanLessThan by blast
then guess U .. note U = this
hence "\<forall>y\<in>U. indicator A y < (1::real)"
unfolding greaterThanLessThan_def by auto
hence "U \<subseteq> -A" by auto
hence "x \<in> interior (-A)" using U interiorI by auto
thus ?thesis using fr interior_complement unfolding frontier_def by auto
qed
next
assume nfr: "x \<notin> frontier A"
hence "x \<in> interior A \<or> x \<in> interior (-A)"
by (auto simp: frontier_def closure_interior)
thus "isCont ((indicator A)::'a \<Rightarrow> real) x"
proof
assume int: "x \<in> interior A"
then obtain U where U: "open U" "x \<in> U" "U \<subseteq> A" unfolding interior_def by auto
hence "\<forall>y\<in>U. indicator A y = (1::real)" unfolding indicator_def by auto
hence "continuous_on U (indicator A)" by (simp add: continuous_on_const indicator_eq_1_iff)
thus ?thesis using U continuous_on_eq_continuous_at by auto
next
assume ext: "x \<in> interior (-A)"
then obtain U where U: "open U" "x \<in> U" "U \<subseteq> -A" unfolding interior_def by auto
then have "continuous_on U (indicator A)"
using continuous_on_topological by (auto simp: subset_iff)
thus ?thesis using U continuous_on_eq_continuous_at by auto
qed
qed
lemma closedin_limpt:
"closedin (subtopology euclidean T) S \<longleftrightarrow> S \<subseteq> T \<and> (\<forall>x. x islimpt S \<and> x \<in> T \<longrightarrow> x \<in> S)"
apply (simp add: closedin_closed, safe)
apply (simp add: closed_limpt islimpt_subset)
apply (rule_tac x="closure S" in exI, simp)
apply (force simp: closure_def)
done
lemma closedin_closed_eq: "closed S \<Longrightarrow> closedin (subtopology euclidean S) T \<longleftrightarrow> closed T \<and> T \<subseteq> S"
by (meson closedin_limpt closed_subset closedin_closed_trans)
lemma connected_closed_set:
"closed S
\<Longrightarrow> connected S \<longleftrightarrow> (\<nexists>A B. closed A \<and> closed B \<and> A \<noteq> {} \<and> B \<noteq> {} \<and> A \<union> B = S \<and> A \<inter> B = {})"
unfolding connected_closedin_eq closedin_closed_eq connected_closedin_eq by blast
text \<open>If a connnected set is written as the union of two nonempty closed sets, then these sets
have to intersect.\<close>
lemma connected_as_closed_union:
assumes "connected C" "C = A \<union> B" "closed A" "closed B" "A \<noteq> {}" "B \<noteq> {}"
shows "A \<inter> B \<noteq> {}"
by (metis assms closed_Un connected_closed_set)
lemma closedin_subset_trans:
"closedin (subtopology euclidean U) S \<Longrightarrow> S \<subseteq> T \<Longrightarrow> T \<subseteq> U \<Longrightarrow>
closedin (subtopology euclidean T) S"
by (meson closedin_limpt subset_iff)
lemma openin_subset_trans:
"openin (subtopology euclidean U) S \<Longrightarrow> S \<subseteq> T \<Longrightarrow> T \<subseteq> U \<Longrightarrow>
openin (subtopology euclidean T) S"
by (auto simp: openin_open)
lemma closedin_compact:
"\<lbrakk>compact S; closedin (subtopology euclidean S) T\<rbrakk> \<Longrightarrow> compact T"
by (metis closedin_closed compact_Int_closed)
lemma closedin_compact_eq:
fixes S :: "'a::t2_space set"
shows
"compact S
\<Longrightarrow> (closedin (subtopology euclidean S) T \<longleftrightarrow>
compact T \<and> T \<subseteq> S)"
by (metis closedin_imp_subset closedin_compact closed_subset compact_imp_closed)
subsection \<open>Closure\<close>
lemma closure_openin_Int_closure:
assumes ope: "openin (subtopology euclidean U) S" and "T \<subseteq> U"
shows "closure(S \<inter> closure T) = closure(S \<inter> T)"
proof
obtain V where "open V" and S: "S = U \<inter> V"
using ope using openin_open by metis
show "closure (S \<inter> closure T) \<subseteq> closure (S \<inter> T)"
proof (clarsimp simp: S)
fix x
assume "x \<in> closure (U \<inter> V \<inter> closure T)"
then have "V \<inter> closure T \<subseteq> A \<Longrightarrow> x \<in> closure A" for A
by (metis closure_mono subsetD inf.coboundedI2 inf_assoc)
then have "x \<in> closure (T \<inter> V)"
by (metis \<open>open V\<close> closure_closure inf_commute open_Int_closure_subset)
then show "x \<in> closure (U \<inter> V \<inter> T)"
by (metis \<open>T \<subseteq> U\<close> inf.absorb_iff2 inf_assoc inf_commute)
qed
next
show "closure (S \<inter> T) \<subseteq> closure (S \<inter> closure T)"
by (meson Int_mono closure_mono closure_subset order_refl)
qed
corollary infinite_openin:
fixes S :: "'a :: t1_space set"
shows "\<lbrakk>openin (subtopology euclidean U) S; x \<in> S; x islimpt U\<rbrakk> \<Longrightarrow> infinite S"
by (clarsimp simp add: openin_open islimpt_eq_acc_point inf_commute)
lemma closure_Int_ballI:
assumes "\<And>U. \<lbrakk>openin (subtopology euclidean S) U; U \<noteq> {}\<rbrakk> \<Longrightarrow> T \<inter> U \<noteq> {}"
shows "S \<subseteq> closure T"
proof (clarsimp simp: closure_iff_nhds_not_empty)
fix x and A and V
assume "x \<in> S" "V \<subseteq> A" "open V" "x \<in> V" "T \<inter> A = {}"
then have "openin (subtopology euclidean S) (A \<inter> V \<inter> S)"
by (auto simp: openin_open intro!: exI[where x="V"])
moreover have "A \<inter> V \<inter> S \<noteq> {}" using \<open>x \<in> V\<close> \<open>V \<subseteq> A\<close> \<open>x \<in> S\<close>
by auto
ultimately have "T \<inter> (A \<inter> V \<inter> S) \<noteq> {}"
by (rule assms)
with \<open>T \<inter> A = {}\<close> show False by auto
qed
subsection \<open>Frontier\<close>
lemma connected_Int_frontier:
"\<lbrakk>connected s; s \<inter> t \<noteq> {}; s - t \<noteq> {}\<rbrakk> \<Longrightarrow> (s \<inter> frontier t \<noteq> {})"
apply (simp add: frontier_interiors connected_openin, safe)
apply (drule_tac x="s \<inter> interior t" in spec, safe)
apply (drule_tac [2] x="s \<inter> interior (-t)" in spec)
apply (auto simp: disjoint_eq_subset_Compl dest: interior_subset [THEN subsetD])
done
subsection \<open>Compactness\<close>
lemma openin_delete:
fixes a :: "'a :: t1_space"
shows "openin (subtopology euclidean u) s
\<Longrightarrow> openin (subtopology euclidean u) (s - {a})"
by (metis Int_Diff open_delete openin_open)
lemma compact_eq_openin_cover:
"compact S \<longleftrightarrow>
(\<forall>C. (\<forall>c\<in>C. openin (subtopology euclidean S) c) \<and> S \<subseteq> \<Union>C \<longrightarrow>
(\<exists>D\<subseteq>C. finite D \<and> S \<subseteq> \<Union>D))"
proof safe
fix C
assume "compact S" and "\<forall>c\<in>C. openin (subtopology euclidean S) c" and "S \<subseteq> \<Union>C"
then have "\<forall>c\<in>{T. open T \<and> S \<inter> T \<in> C}. open c" and "S \<subseteq> \<Union>{T. open T \<and> S \<inter> T \<in> C}"
unfolding openin_open by force+
with \<open>compact S\<close> obtain D where "D \<subseteq> {T. open T \<and> S \<inter> T \<in> C}" and "finite D" and "S \<subseteq> \<Union>D"
by (meson compactE)
then have "image (\<lambda>T. S \<inter> T) D \<subseteq> C \<and> finite (image (\<lambda>T. S \<inter> T) D) \<and> S \<subseteq> \<Union>(image (\<lambda>T. S \<inter> T) D)"
by auto
then show "\<exists>D\<subseteq>C. finite D \<and> S \<subseteq> \<Union>D" ..
next
assume 1: "\<forall>C. (\<forall>c\<in>C. openin (subtopology euclidean S) c) \<and> S \<subseteq> \<Union>C \<longrightarrow>
(\<exists>D\<subseteq>C. finite D \<and> S \<subseteq> \<Union>D)"
show "compact S"
proof (rule compactI)
fix C
let ?C = "image (\<lambda>T. S \<inter> T) C"
assume "\<forall>t\<in>C. open t" and "S \<subseteq> \<Union>C"
then have "(\<forall>c\<in>?C. openin (subtopology euclidean S) c) \<and> S \<subseteq> \<Union>?C"
unfolding openin_open by auto
with 1 obtain D where "D \<subseteq> ?C" and "finite D" and "S \<subseteq> \<Union>D"
by metis
let ?D = "inv_into C (\<lambda>T. S \<inter> T) ` D"
have "?D \<subseteq> C \<and> finite ?D \<and> S \<subseteq> \<Union>?D"
proof (intro conjI)
from \<open>D \<subseteq> ?C\<close> show "?D \<subseteq> C"
by (fast intro: inv_into_into)
from \<open>finite D\<close> show "finite ?D"
by (rule finite_imageI)
from \<open>S \<subseteq> \<Union>D\<close> show "S \<subseteq> \<Union>?D"
apply (rule subset_trans, clarsimp)
apply (frule subsetD [OF \<open>D \<subseteq> ?C\<close>, THEN f_inv_into_f])
apply (erule rev_bexI, fast)
done
qed
then show "\<exists>D\<subseteq>C. finite D \<and> S \<subseteq> \<Union>D" ..
qed
qed
subsection \<open>Continuity\<close>
lemma interior_image_subset:
assumes "inj f" "\<And>x. continuous (at x) f"
shows "interior (f ` S) \<subseteq> f ` (interior S)"
proof
fix x assume "x \<in> interior (f ` S)"
then obtain T where as: "open T" "x \<in> T" "T \<subseteq> f ` S" ..
then have "x \<in> f ` S" by auto
then obtain y where y: "y \<in> S" "x = f y" by auto
have "open (f -` T)"
using assms \<open>open T\<close> by (simp add: continuous_at_imp_continuous_on open_vimage)
moreover have "y \<in> vimage f T"
using \<open>x = f y\<close> \<open>x \<in> T\<close> by simp
moreover have "vimage f T \<subseteq> S"
using \<open>T \<subseteq> image f S\<close> \<open>inj f\<close> unfolding inj_on_def subset_eq by auto
ultimately have "y \<in> interior S" ..
with \<open>x = f y\<close> show "x \<in> f ` interior S" ..
qed
subsection%unimportant \<open>Equality of continuous functions on closure and related results\<close>
lemma continuous_closedin_preimage_constant:
fixes f :: "_ \<Rightarrow> 'b::t1_space"
shows "continuous_on S f \<Longrightarrow> closedin (subtopology euclidean S) {x \<in> S. f x = a}"
using continuous_closedin_preimage[of S f "{a}"] by (simp add: vimage_def Collect_conj_eq)
lemma continuous_closed_preimage_constant:
fixes f :: "_ \<Rightarrow> 'b::t1_space"
shows "continuous_on S f \<Longrightarrow> closed S \<Longrightarrow> closed {x \<in> S. f x = a}"
using continuous_closed_preimage[of S f "{a}"] by (simp add: vimage_def Collect_conj_eq)
lemma continuous_constant_on_closure:
fixes f :: "_ \<Rightarrow> 'b::t1_space"
assumes "continuous_on (closure S) f"
and "\<And>x. x \<in> S \<Longrightarrow> f x = a"
and "x \<in> closure S"
shows "f x = a"
using continuous_closed_preimage_constant[of "closure S" f a]
assms closure_minimal[of S "{x \<in> closure S. f x = a}"] closure_subset
unfolding subset_eq
by auto
lemma image_closure_subset:
assumes contf: "continuous_on (closure S) f"
and "closed T"
and "(f ` S) \<subseteq> T"
shows "f ` (closure S) \<subseteq> T"
proof -
have "S \<subseteq> {x \<in> closure S. f x \<in> T}"
using assms(3) closure_subset by auto
moreover have "closed (closure S \<inter> f -` T)"
using continuous_closed_preimage[OF contf] \<open>closed T\<close> by auto
ultimately have "closure S = (closure S \<inter> f -` T)"
using closure_minimal[of S "(closure S \<inter> f -` T)"] by auto
then show ?thesis by auto
qed
subsection%unimportant \<open>A function constant on a set\<close>
definition constant_on (infixl "(constant'_on)" 50)
where "f constant_on A \<equiv> \<exists>y. \<forall>x\<in>A. f x = y"
lemma constant_on_subset: "\<lbrakk>f constant_on A; B \<subseteq> A\<rbrakk> \<Longrightarrow> f constant_on B"
unfolding constant_on_def by blast
lemma injective_not_constant:
fixes S :: "'a::{perfect_space} set"
shows "\<lbrakk>open S; inj_on f S; f constant_on S\<rbrakk> \<Longrightarrow> S = {}"
unfolding constant_on_def
by (metis equals0I inj_on_contraD islimpt_UNIV islimpt_def)
lemma constant_on_closureI:
fixes f :: "_ \<Rightarrow> 'b::t1_space"
assumes cof: "f constant_on S" and contf: "continuous_on (closure S) f"
shows "f constant_on (closure S)"
using continuous_constant_on_closure [OF contf] cof unfolding constant_on_def
by metis
subsection%unimportant \<open>Continuity relative to a union.\<close>
lemma continuous_on_Un_local:
"\<lbrakk>closedin (subtopology euclidean (s \<union> t)) s; closedin (subtopology euclidean (s \<union> t)) t;
continuous_on s f; continuous_on t f\<rbrakk>
\<Longrightarrow> continuous_on (s \<union> t) f"
unfolding continuous_on closedin_limpt
by (metis Lim_trivial_limit Lim_within_union Un_iff trivial_limit_within)
lemma continuous_on_cases_local:
"\<lbrakk>closedin (subtopology euclidean (s \<union> t)) s; closedin (subtopology euclidean (s \<union> t)) t;
continuous_on s f; continuous_on t g;
\<And>x. \<lbrakk>x \<in> s \<and> \<not>P x \<or> x \<in> t \<and> P x\<rbrakk> \<Longrightarrow> f x = g x\<rbrakk>
\<Longrightarrow> continuous_on (s \<union> t) (\<lambda>x. if P x then f x else g x)"
by (rule continuous_on_Un_local) (auto intro: continuous_on_eq)
lemma continuous_on_cases_le:
fixes h :: "'a :: topological_space \<Rightarrow> real"
assumes "continuous_on {t \<in> s. h t \<le> a} f"
and "continuous_on {t \<in> s. a \<le> h t} g"
and h: "continuous_on s h"
and "\<And>t. \<lbrakk>t \<in> s; h t = a\<rbrakk> \<Longrightarrow> f t = g t"
shows "continuous_on s (\<lambda>t. if h t \<le> a then f(t) else g(t))"
proof -
have s: "s = (s \<inter> h -` atMost a) \<union> (s \<inter> h -` atLeast a)"
by force
have 1: "closedin (subtopology euclidean s) (s \<inter> h -` atMost a)"
by (rule continuous_closedin_preimage [OF h closed_atMost])
have 2: "closedin (subtopology euclidean s) (s \<inter> h -` atLeast a)"
by (rule continuous_closedin_preimage [OF h closed_atLeast])
have eq: "s \<inter> h -` {..a} = {t \<in> s. h t \<le> a}" "s \<inter> h -` {a..} = {t \<in> s. a \<le> h t}"
by auto
show ?thesis
apply (rule continuous_on_subset [of s, OF _ order_refl])
apply (subst s)
apply (rule continuous_on_cases_local)
using 1 2 s assms apply (auto simp: eq)
done
qed
lemma continuous_on_cases_1:
fixes s :: "real set"
assumes "continuous_on {t \<in> s. t \<le> a} f"
and "continuous_on {t \<in> s. a \<le> t} g"
and "a \<in> s \<Longrightarrow> f a = g a"
shows "continuous_on s (\<lambda>t. if t \<le> a then f(t) else g(t))"
using assms
by (auto simp: continuous_on_id intro: continuous_on_cases_le [where h = id, simplified])
subsection%unimportant\<open>Inverse function property for open/closed maps\<close>
lemma continuous_on_inverse_open_map:
assumes contf: "continuous_on S f"
and imf: "f ` S = T"
and injf: "\<And>x. x \<in> S \<Longrightarrow> g (f x) = x"
and oo: "\<And>U. openin (subtopology euclidean S) U \<Longrightarrow> openin (subtopology euclidean T) (f ` U)"
shows "continuous_on T g"
proof -
from imf injf have gTS: "g ` T = S"
by force
from imf injf have fU: "U \<subseteq> S \<Longrightarrow> (f ` U) = T \<inter> g -` U" for U
by force
show ?thesis
by (simp add: continuous_on_open [of T g] gTS) (metis openin_imp_subset fU oo)
qed
lemma continuous_on_inverse_closed_map:
assumes contf: "continuous_on S f"
and imf: "f ` S = T"
and injf: "\<And>x. x \<in> S \<Longrightarrow> g(f x) = x"
and oo: "\<And>U. closedin (subtopology euclidean S) U \<Longrightarrow> closedin (subtopology euclidean T) (f ` U)"
shows "continuous_on T g"
proof -
from imf injf have gTS: "g ` T = S"
by force
from imf injf have fU: "U \<subseteq> S \<Longrightarrow> (f ` U) = T \<inter> g -` U" for U
by force
show ?thesis
by (simp add: continuous_on_closed [of T g] gTS) (metis closedin_imp_subset fU oo)
qed
lemma homeomorphism_injective_open_map:
assumes contf: "continuous_on S f"
and imf: "f ` S = T"
and injf: "inj_on f S"
and oo: "\<And>U. openin (subtopology euclidean S) U \<Longrightarrow> openin (subtopology euclidean T) (f ` U)"
obtains g where "homeomorphism S T f g"
proof
have "continuous_on T (inv_into S f)"
by (metis contf continuous_on_inverse_open_map imf injf inv_into_f_f oo)
with imf injf contf show "homeomorphism S T f (inv_into S f)"
by (auto simp: homeomorphism_def)
qed
lemma homeomorphism_injective_closed_map:
assumes contf: "continuous_on S f"
and imf: "f ` S = T"
and injf: "inj_on f S"
and oo: "\<And>U. closedin (subtopology euclidean S) U \<Longrightarrow> closedin (subtopology euclidean T) (f ` U)"
obtains g where "homeomorphism S T f g"
proof
have "continuous_on T (inv_into S f)"
by (metis contf continuous_on_inverse_closed_map imf injf inv_into_f_f oo)
with imf injf contf show "homeomorphism S T f (inv_into S f)"
by (auto simp: homeomorphism_def)
qed
lemma homeomorphism_imp_open_map:
assumes hom: "homeomorphism S T f g"
and oo: "openin (subtopology euclidean S) U"
shows "openin (subtopology euclidean T) (f ` U)"
proof -
from hom oo have [simp]: "f ` U = T \<inter> g -` U"
using openin_subset by (fastforce simp: homeomorphism_def rev_image_eqI)
from hom have "continuous_on T g"
unfolding homeomorphism_def by blast
moreover have "g ` T = S"
by (metis hom homeomorphism_def)
ultimately show ?thesis
by (simp add: continuous_on_open oo)
qed
lemma homeomorphism_imp_closed_map:
assumes hom: "homeomorphism S T f g"
and oo: "closedin (subtopology euclidean S) U"
shows "closedin (subtopology euclidean T) (f ` U)"
proof -
from hom oo have [simp]: "f ` U = T \<inter> g -` U"
using closedin_subset by (fastforce simp: homeomorphism_def rev_image_eqI)
from hom have "continuous_on T g"
unfolding homeomorphism_def by blast
moreover have "g ` T = S"
by (metis hom homeomorphism_def)
ultimately show ?thesis
by (simp add: continuous_on_closed oo)
qed
subsection%unimportant \<open>Seperability\<close>
lemma subset_second_countable:
obtains \<B> :: "'a:: second_countable_topology set set"
where "countable \<B>"
"{} \<notin> \<B>"
"\<And>C. C \<in> \<B> \<Longrightarrow> openin(subtopology euclidean S) C"
"\<And>T. openin(subtopology euclidean S) T \<Longrightarrow> \<exists>\<U>. \<U> \<subseteq> \<B> \<and> T = \<Union>\<U>"
proof -
obtain \<B> :: "'a set set"
where "countable \<B>"
and opeB: "\<And>C. C \<in> \<B> \<Longrightarrow> openin(subtopology euclidean S) C"
and \<B>: "\<And>T. openin(subtopology euclidean S) T \<Longrightarrow> \<exists>\<U>. \<U> \<subseteq> \<B> \<and> T = \<Union>\<U>"
proof -
obtain \<C> :: "'a set set"
where "countable \<C>" and ope: "\<And>C. C \<in> \<C> \<Longrightarrow> open C"
and \<C>: "\<And>S. open S \<Longrightarrow> \<exists>U. U \<subseteq> \<C> \<and> S = \<Union>U"
by (metis univ_second_countable that)
show ?thesis
proof
show "countable ((\<lambda>C. S \<inter> C) ` \<C>)"
by (simp add: \<open>countable \<C>\<close>)
show "\<And>C. C \<in> (\<inter>) S ` \<C> \<Longrightarrow> openin (subtopology euclidean S) C"
using ope by auto
show "\<And>T. openin (subtopology euclidean S) T \<Longrightarrow> \<exists>\<U>\<subseteq>(\<inter>) S ` \<C>. T = \<Union>\<U>"
by (metis \<C> image_mono inf_Sup openin_open)
qed
qed
show ?thesis
proof
show "countable (\<B> - {{}})"
using \<open>countable \<B>\<close> by blast
show "\<And>C. \<lbrakk>C \<in> \<B> - {{}}\<rbrakk> \<Longrightarrow> openin (subtopology euclidean S) C"
by (simp add: \<open>\<And>C. C \<in> \<B> \<Longrightarrow> openin (subtopology euclidean S) C\<close>)
show "\<exists>\<U>\<subseteq>\<B> - {{}}. T = \<Union>\<U>" if "openin (subtopology euclidean S) T" for T
using \<B> [OF that]
apply clarify
apply (rule_tac x="\<U> - {{}}" in exI, auto)
done
qed auto
qed
lemma Lindelof_openin:
fixes \<F> :: "'a::second_countable_topology set set"
assumes "\<And>S. S \<in> \<F> \<Longrightarrow> openin (subtopology euclidean U) S"
obtains \<F>' where "\<F>' \<subseteq> \<F>" "countable \<F>'" "\<Union>\<F>' = \<Union>\<F>"
proof -
have "\<And>S. S \<in> \<F> \<Longrightarrow> \<exists>T. open T \<and> S = U \<inter> T"
using assms by (simp add: openin_open)
then obtain tf where tf: "\<And>S. S \<in> \<F> \<Longrightarrow> open (tf S) \<and> (S = U \<inter> tf S)"
by metis
have [simp]: "\<And>\<F>'. \<F>' \<subseteq> \<F> \<Longrightarrow> \<Union>\<F>' = U \<inter> \<Union>(tf ` \<F>')"
using tf by fastforce
obtain \<G> where "countable \<G> \<and> \<G> \<subseteq> tf ` \<F>" "\<Union>\<G> = \<Union>(tf ` \<F>)"
using tf by (force intro: Lindelof [of "tf ` \<F>"])
then obtain \<F>' where \<F>': "\<F>' \<subseteq> \<F>" "countable \<F>'" "\<Union>\<F>' = \<Union>\<F>"
by (clarsimp simp add: countable_subset_image)
then show ?thesis ..
qed
subsection%unimportant\<open>Closed Maps\<close>
lemma continuous_imp_closed_map:
fixes f :: "'a::t2_space \<Rightarrow> 'b::t2_space"
assumes "closedin (subtopology euclidean S) U"
"continuous_on S f" "f ` S = T" "compact S"
shows "closedin (subtopology euclidean T) (f ` U)"
by (metis assms closedin_compact_eq compact_continuous_image continuous_on_subset subset_image_iff)
lemma closed_map_restrict:
assumes cloU: "closedin (subtopology euclidean (S \<inter> f -` T')) U"
and cc: "\<And>U. closedin (subtopology euclidean S) U \<Longrightarrow> closedin (subtopology euclidean T) (f ` U)"
and "T' \<subseteq> T"
shows "closedin (subtopology euclidean T') (f ` U)"
proof -
obtain V where "closed V" "U = S \<inter> f -` T' \<inter> V"
using cloU by (auto simp: closedin_closed)
with cc [of "S \<inter> V"] \<open>T' \<subseteq> T\<close> show ?thesis
by (fastforce simp add: closedin_closed)
qed
subsection%unimportant\<open>Open Maps\<close>
lemma open_map_restrict:
assumes opeU: "openin (subtopology euclidean (S \<inter> f -` T')) U"
and oo: "\<And>U. openin (subtopology euclidean S) U \<Longrightarrow> openin (subtopology euclidean T) (f ` U)"
and "T' \<subseteq> T"
shows "openin (subtopology euclidean T') (f ` U)"
proof -
obtain V where "open V" "U = S \<inter> f -` T' \<inter> V"
using opeU by (auto simp: openin_open)
with oo [of "S \<inter> V"] \<open>T' \<subseteq> T\<close> show ?thesis
by (fastforce simp add: openin_open)
qed
subsection%unimportant\<open>Quotient maps\<close>
lemma quotient_map_imp_continuous_open:
assumes T: "f ` S \<subseteq> T"
and ope: "\<And>U. U \<subseteq> T
\<Longrightarrow> (openin (subtopology euclidean S) (S \<inter> f -` U) \<longleftrightarrow>
openin (subtopology euclidean T) U)"
shows "continuous_on S f"
proof -
have [simp]: "S \<inter> f -` f ` S = S" by auto
show ?thesis
using ope [OF T]
apply (simp add: continuous_on_open)
by (meson ope openin_imp_subset openin_trans)
qed
lemma quotient_map_imp_continuous_closed:
assumes T: "f ` S \<subseteq> T"
and ope: "\<And>U. U \<subseteq> T
\<Longrightarrow> (closedin (subtopology euclidean S) (S \<inter> f -` U) \<longleftrightarrow>
closedin (subtopology euclidean T) U)"
shows "continuous_on S f"
proof -
have [simp]: "S \<inter> f -` f ` S = S" by auto
show ?thesis
using ope [OF T]
apply (simp add: continuous_on_closed)
by (metis (no_types, lifting) ope closedin_imp_subset closedin_trans)
qed
lemma open_map_imp_quotient_map:
assumes contf: "continuous_on S f"
and T: "T \<subseteq> f ` S"
and ope: "\<And>T. openin (subtopology euclidean S) T
\<Longrightarrow> openin (subtopology euclidean (f ` S)) (f ` T)"
shows "openin (subtopology euclidean S) (S \<inter> f -` T) =
openin (subtopology euclidean (f ` S)) T"
proof -
have "T = f ` (S \<inter> f -` T)"
using T by blast
then show ?thesis
using "ope" contf continuous_on_open by metis
qed
lemma closed_map_imp_quotient_map:
assumes contf: "continuous_on S f"
and T: "T \<subseteq> f ` S"
and ope: "\<And>T. closedin (subtopology euclidean S) T
\<Longrightarrow> closedin (subtopology euclidean (f ` S)) (f ` T)"
shows "openin (subtopology euclidean S) (S \<inter> f -` T) \<longleftrightarrow>
openin (subtopology euclidean (f ` S)) T"
(is "?lhs = ?rhs")
proof
assume ?lhs
then have *: "closedin (subtopology euclidean S) (S - (S \<inter> f -` T))"
using closedin_diff by fastforce
have [simp]: "(f ` S - f ` (S - (S \<inter> f -` T))) = T"
using T by blast
show ?rhs
using ope [OF *, unfolded closedin_def] by auto
next
assume ?rhs
with contf show ?lhs
by (auto simp: continuous_on_open)
qed
lemma continuous_right_inverse_imp_quotient_map:
assumes contf: "continuous_on S f" and imf: "f ` S \<subseteq> T"
and contg: "continuous_on T g" and img: "g ` T \<subseteq> S"
and fg [simp]: "\<And>y. y \<in> T \<Longrightarrow> f(g y) = y"
and U: "U \<subseteq> T"
shows "openin (subtopology euclidean S) (S \<inter> f -` U) \<longleftrightarrow>
openin (subtopology euclidean T) U"
(is "?lhs = ?rhs")
proof -
have f: "\<And>Z. openin (subtopology euclidean (f ` S)) Z \<Longrightarrow>
openin (subtopology euclidean S) (S \<inter> f -` Z)"
and g: "\<And>Z. openin (subtopology euclidean (g ` T)) Z \<Longrightarrow>
openin (subtopology euclidean T) (T \<inter> g -` Z)"
using contf contg by (auto simp: continuous_on_open)
show ?thesis
proof
have "T \<inter> g -` (g ` T \<inter> (S \<inter> f -` U)) = {x \<in> T. f (g x) \<in> U}"
using imf img by blast
also have "... = U"
using U by auto
finally have eq: "T \<inter> g -` (g ` T \<inter> (S \<inter> f -` U)) = U" .
assume ?lhs
then have *: "openin (subtopology euclidean (g ` T)) (g ` T \<inter> (S \<inter> f -` U))"
by (meson img openin_Int openin_subtopology_Int_subset openin_subtopology_self)
show ?rhs
using g [OF *] eq by auto
next
assume rhs: ?rhs
show ?lhs
by (metis f fg image_eqI image_subset_iff imf img openin_subopen openin_subtopology_self openin_trans rhs)
qed
qed
lemma continuous_left_inverse_imp_quotient_map:
assumes "continuous_on S f"
and "continuous_on (f ` S) g"
and "\<And>x. x \<in> S \<Longrightarrow> g(f x) = x"
and "U \<subseteq> f ` S"
shows "openin (subtopology euclidean S) (S \<inter> f -` U) \<longleftrightarrow>
openin (subtopology euclidean (f ` S)) U"
apply (rule continuous_right_inverse_imp_quotient_map)
using assms apply force+
done
lemma continuous_imp_quotient_map:
fixes f :: "'a::t2_space \<Rightarrow> 'b::t2_space"
assumes "continuous_on S f" "f ` S = T" "compact S" "U \<subseteq> T"
shows "openin (subtopology euclidean S) (S \<inter> f -` U) \<longleftrightarrow>
openin (subtopology euclidean T) U"
by (metis (no_types, lifting) assms closed_map_imp_quotient_map continuous_imp_closed_map)
subsection%unimportant\<open>Pasting functions together\<close>
text\<open>on open sets\<close>
lemma pasting_lemma:
fixes f :: "'i \<Rightarrow> 'a::topological_space \<Rightarrow> 'b::topological_space"
assumes clo: "\<And>i. i \<in> I \<Longrightarrow> openin (subtopology euclidean S) (T i)"
and cont: "\<And>i. i \<in> I \<Longrightarrow> continuous_on (T i) (f i)"
and f: "\<And>i j x. \<lbrakk>i \<in> I; j \<in> I; x \<in> S \<inter> T i \<inter> T j\<rbrakk> \<Longrightarrow> f i x = f j x"
and g: "\<And>x. x \<in> S \<Longrightarrow> \<exists>j. j \<in> I \<and> x \<in> T j \<and> g x = f j x"
shows "continuous_on S g"
proof (clarsimp simp: continuous_openin_preimage_eq)
fix U :: "'b set"
assume "open U"
have S: "\<And>i. i \<in> I \<Longrightarrow> (T i) \<subseteq> S"
using clo openin_imp_subset by blast
have *: "(S \<inter> g -` U) = (\<Union>i \<in> I. T i \<inter> f i -` U)"
using S f g by fastforce
show "openin (subtopology euclidean S) (S \<inter> g -` U)"
apply (subst *)
apply (rule openin_Union, clarify)
using \<open>open U\<close> clo cont continuous_openin_preimage_gen openin_trans by blast
qed
lemma pasting_lemma_exists:
fixes f :: "'i \<Rightarrow> 'a::topological_space \<Rightarrow> 'b::topological_space"
assumes S: "S \<subseteq> (\<Union>i \<in> I. T i)"
and clo: "\<And>i. i \<in> I \<Longrightarrow> openin (subtopology euclidean S) (T i)"
and cont: "\<And>i. i \<in> I \<Longrightarrow> continuous_on (T i) (f i)"
and f: "\<And>i j x. \<lbrakk>i \<in> I; j \<in> I; x \<in> S \<inter> T i \<inter> T j\<rbrakk> \<Longrightarrow> f i x = f j x"
obtains g where "continuous_on S g" "\<And>x i. \<lbrakk>i \<in> I; x \<in> S \<inter> T i\<rbrakk> \<Longrightarrow> g x = f i x"
proof
show "continuous_on S (\<lambda>x. f (SOME i. i \<in> I \<and> x \<in> T i) x)"
apply (rule pasting_lemma [OF clo cont])
apply (blast intro: f)+
apply (metis (mono_tags, lifting) S UN_iff subsetCE someI)
done
next
fix x i
assume "i \<in> I" "x \<in> S \<inter> T i"
then show "f (SOME i. i \<in> I \<and> x \<in> T i) x = f i x"
by (metis (no_types, lifting) IntD2 IntI f someI_ex)
qed
text\<open>Likewise on closed sets, with a finiteness assumption\<close>
lemma pasting_lemma_closed:
fixes f :: "'i \<Rightarrow> 'a::topological_space \<Rightarrow> 'b::topological_space"
assumes "finite I"
and clo: "\<And>i. i \<in> I \<Longrightarrow> closedin (subtopology euclidean S) (T i)"
and cont: "\<And>i. i \<in> I \<Longrightarrow> continuous_on (T i) (f i)"
and f: "\<And>i j x. \<lbrakk>i \<in> I; j \<in> I; x \<in> S \<inter> T i \<inter> T j\<rbrakk> \<Longrightarrow> f i x = f j x"
and g: "\<And>x. x \<in> S \<Longrightarrow> \<exists>j. j \<in> I \<and> x \<in> T j \<and> g x = f j x"
shows "continuous_on S g"
proof (clarsimp simp: continuous_closedin_preimage_eq)
fix U :: "'b set"
assume "closed U"
have S: "\<And>i. i \<in> I \<Longrightarrow> (T i) \<subseteq> S"
using clo closedin_imp_subset by blast
have *: "(S \<inter> g -` U) = (\<Union>i \<in> I. T i \<inter> f i -` U)"
using S f g by fastforce
show "closedin (subtopology euclidean S) (S \<inter> g -` U)"
apply (subst *)
apply (rule closedin_Union)
using \<open>finite I\<close> apply simp
apply (blast intro: \<open>closed U\<close> continuous_closedin_preimage cont clo closedin_trans)
done
qed
lemma pasting_lemma_exists_closed:
fixes f :: "'i \<Rightarrow> 'a::topological_space \<Rightarrow> 'b::topological_space"
assumes "finite I"
and S: "S \<subseteq> (\<Union>i \<in> I. T i)"
and clo: "\<And>i. i \<in> I \<Longrightarrow> closedin (subtopology euclidean S) (T i)"
and cont: "\<And>i. i \<in> I \<Longrightarrow> continuous_on (T i) (f i)"
and f: "\<And>i j x. \<lbrakk>i \<in> I; j \<in> I; x \<in> S \<inter> T i \<inter> T j\<rbrakk> \<Longrightarrow> f i x = f j x"
obtains g where "continuous_on S g" "\<And>x i. \<lbrakk>i \<in> I; x \<in> S \<inter> T i\<rbrakk> \<Longrightarrow> g x = f i x"
proof
show "continuous_on S (\<lambda>x. f (SOME i. i \<in> I \<and> x \<in> T i) x)"
apply (rule pasting_lemma_closed [OF \<open>finite I\<close> clo cont])
apply (blast intro: f)+
apply (metis (mono_tags, lifting) S UN_iff subsetCE someI)
done
next
fix x i
assume "i \<in> I" "x \<in> S \<inter> T i"
then show "f (SOME i. i \<in> I \<and> x \<in> T i) x = f i x"
by (metis (no_types, lifting) IntD2 IntI f someI_ex)
qed
end