author  immler 
Tue, 18 Mar 2014 10:12:57 +0100  
changeset 56188  0268784f60da 
parent 56166  9a241bc276cd 
child 56189  c4daa97ac57a 
permissions  rwrr 
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(* title: HOL/Library/Topology_Euclidian_Space.thy 
33175  2 
Author: Amine Chaieb, University of Cambridge 
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Author: Robert Himmelmann, TU Muenchen 

44075  4 
Author: Brian Huffman, Portland State University 
33175  5 
*) 
6 

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header {* Elementary topology in Euclidean space. *} 

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theory Topology_Euclidean_Space 

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imports 
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Complex_Main 
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"~~/src/HOL/Library/Countable_Set" 
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"~~/src/HOL/Library/FuncSet" 
50938  14 
Linear_Algebra 
50087  15 
Norm_Arith 
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begin 

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50972  18 
lemma dist_0_norm: 
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fixes x :: "'a::real_normed_vector" 

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shows "dist 0 x = norm x" 

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unfolding dist_norm by simp 

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lemma dist_double: "dist x y < d / 2 \<Longrightarrow> dist x z < d / 2 \<Longrightarrow> dist y z < d" 
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using dist_triangle[of y z x] by (simp add: dist_commute) 
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(* LEGACY *) 
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lemma lim_subseq: "subseq r \<Longrightarrow> s > l \<Longrightarrow> (s \<circ> r) > l" 
50972  28 
by (rule LIMSEQ_subseq_LIMSEQ) 
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lemma countable_PiE: 
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"finite I \<Longrightarrow> (\<And>i. i \<in> I \<Longrightarrow> countable (F i)) \<Longrightarrow> countable (PiE I F)" 
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by (induct I arbitrary: F rule: finite_induct) (auto simp: PiE_insert_eq) 
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lemma Lim_within_open: 
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fixes f :: "'a::topological_space \<Rightarrow> 'b::topological_space" 
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shows "a \<in> S \<Longrightarrow> open S \<Longrightarrow> (f > l)(at a within S) \<longleftrightarrow> (f > l)(at a)" 
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by (fact tendsto_within_open) 
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lemma continuous_on_union: 
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"closed s \<Longrightarrow> closed t \<Longrightarrow> continuous_on s f \<Longrightarrow> continuous_on t f \<Longrightarrow> continuous_on (s \<union> t) f" 
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by (fact continuous_on_closed_Un) 
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lemma continuous_on_cases: 
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"closed s \<Longrightarrow> closed t \<Longrightarrow> continuous_on s f \<Longrightarrow> continuous_on t g \<Longrightarrow> 
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\<forall>x. (x\<in>s \<and> \<not> P x) \<or> (x \<in> t \<and> P x) \<longrightarrow> f x = g x \<Longrightarrow> 
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continuous_on (s \<union> t) (\<lambda>x. if P x then f x else g x)" 
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by (rule continuous_on_If) auto 
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53255  49 

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subsection {* Topological Basis *} 
51 

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context topological_space 

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begin 

54 

53291  55 
definition "topological_basis B \<longleftrightarrow> 
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(\<forall>b\<in>B. open b) \<and> (\<forall>x. open x \<longrightarrow> (\<exists>B'. B' \<subseteq> B \<and> \<Union>B' = x))" 

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lemma topological_basis: 
53291  59 
"topological_basis B \<longleftrightarrow> (\<forall>x. open x \<longleftrightarrow> (\<exists>B'. B' \<subseteq> B \<and> \<Union>B' = x))" 
50998  60 
unfolding topological_basis_def 
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apply safe 

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apply fastforce 

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apply fastforce 

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apply (erule_tac x="x" in allE) 

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apply simp 

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apply (rule_tac x="{x}" in exI) 

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apply auto 

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done 

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lemma topological_basis_iff: 
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assumes "\<And>B'. B' \<in> B \<Longrightarrow> open B'" 

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shows "topological_basis B \<longleftrightarrow> (\<forall>O'. open O' \<longrightarrow> (\<forall>x\<in>O'. \<exists>B'\<in>B. x \<in> B' \<and> B' \<subseteq> O'))" 

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(is "_ \<longleftrightarrow> ?rhs") 

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proof safe 

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fix O' and x::'a 

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assume H: "topological_basis B" "open O'" "x \<in> O'" 

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then have "(\<exists>B'\<subseteq>B. \<Union>B' = O')" by (simp add: topological_basis_def) 
50087  78 
then obtain B' where "B' \<subseteq> B" "O' = \<Union>B'" by auto 
53282  79 
then show "\<exists>B'\<in>B. x \<in> B' \<and> B' \<subseteq> O'" using H by auto 
50087  80 
next 
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assume H: ?rhs 

53282  82 
show "topological_basis B" 
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using assms unfolding topological_basis_def 

50087  84 
proof safe 
53640  85 
fix O' :: "'a set" 
53282  86 
assume "open O'" 
50087  87 
with H obtain f where "\<forall>x\<in>O'. f x \<in> B \<and> x \<in> f x \<and> f x \<subseteq> O'" 
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by (force intro: bchoice simp: Bex_def) 

53282  89 
then show "\<exists>B'\<subseteq>B. \<Union>B' = O'" 
50087  90 
by (auto intro: exI[where x="{f x x. x \<in> O'}"]) 
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qed 

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qed 

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lemma topological_basisI: 

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assumes "\<And>B'. B' \<in> B \<Longrightarrow> open B'" 

53282  96 
and "\<And>O' x. open O' \<Longrightarrow> x \<in> O' \<Longrightarrow> \<exists>B'\<in>B. x \<in> B' \<and> B' \<subseteq> O'" 
50087  97 
shows "topological_basis B" 
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using assms by (subst topological_basis_iff) auto 

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lemma topological_basisE: 

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fixes O' 

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assumes "topological_basis B" 

53282  103 
and "open O'" 
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and "x \<in> O'" 

50087  105 
obtains B' where "B' \<in> B" "x \<in> B'" "B' \<subseteq> O'" 
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proof atomize_elim 

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from assms have "\<And>B'. B'\<in>B \<Longrightarrow> open B'" 
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by (simp add: topological_basis_def) 

50087  109 
with topological_basis_iff assms 
53282  110 
show "\<exists>B'. B' \<in> B \<and> x \<in> B' \<and> B' \<subseteq> O'" 
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using assms by (simp add: Bex_def) 

50087  112 
qed 
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lemma topological_basis_open: 
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assumes "topological_basis B" 
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and "X \<in> B" 
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shows "open X" 
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using assms by (simp add: topological_basis_def) 
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lemma topological_basis_imp_subbasis: 
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assumes B: "topological_basis B" 
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shows "open = generate_topology B" 

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proof (intro ext iffI) 
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fix S :: "'a set" 
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assume "open S" 

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with B obtain B' where "B' \<subseteq> B" "S = \<Union>B'" 
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unfolding topological_basis_def by blast 
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then show "generate_topology B S" 
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by (auto intro: generate_topology.intros dest: topological_basis_open) 
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next 
53255  131 
fix S :: "'a set" 
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assume "generate_topology B S" 

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then show "open S" 

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by induct (auto dest: topological_basis_open[OF B]) 
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qed 
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lemma basis_dense: 
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fixes B :: "'a set set" 
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and f :: "'a set \<Rightarrow> 'a" 

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assumes "topological_basis B" 
53255  141 
and choosefrom_basis: "\<And>B'. B' \<noteq> {} \<Longrightarrow> f B' \<in> B'" 
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shows "\<forall>X. open X \<longrightarrow> X \<noteq> {} \<longrightarrow> (\<exists>B' \<in> B. f B' \<in> X)" 
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proof (intro allI impI) 
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fix X :: "'a set" 
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assume "open X" and "X \<noteq> {}" 

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from topological_basisE[OF `topological_basis B` `open X` choosefrom_basis[OF `X \<noteq> {}`]] 
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obtain B' where "B' \<in> B" "f X \<in> B'" "B' \<subseteq> X" . 
53255  148 
then show "\<exists>B'\<in>B. f B' \<in> X" 
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by (auto intro!: choosefrom_basis) 

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qed 
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50087  152 
end 
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lemma topological_basis_prod: 
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assumes A: "topological_basis A" 
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and B: "topological_basis B" 

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shows "topological_basis ((\<lambda>(a, b). a \<times> b) ` (A \<times> B))" 
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unfolding topological_basis_def 
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proof (safe, simp_all del: ex_simps add: subset_image_iff ex_simps(1)[symmetric]) 
53255  160 
fix S :: "('a \<times> 'b) set" 
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assume "open S" 

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then show "\<exists>X\<subseteq>A \<times> B. (\<Union>(a,b)\<in>X. a \<times> b) = S" 
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proof (safe intro!: exI[of _ "{x\<in>A \<times> B. fst x \<times> snd x \<subseteq> S}"]) 
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fix x y 
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assume "(x, y) \<in> S" 

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from open_prod_elim[OF `open S` this] 
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obtain a b where a: "open a""x \<in> a" and b: "open b" "y \<in> b" and "a \<times> b \<subseteq> S" 
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by (metis mem_Sigma_iff) 
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moreover 
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from A a obtain A0 where "A0 \<in> A" "x \<in> A0" "A0 \<subseteq> a" 

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by (rule topological_basisE) 

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moreover 

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from B b obtain B0 where "B0 \<in> B" "y \<in> B0" "B0 \<subseteq> b" 

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by (rule topological_basisE) 

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ultimately show "(x, y) \<in> (\<Union>(a, b)\<in>{X \<in> A \<times> B. fst X \<times> snd X \<subseteq> S}. a \<times> b)" 
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by (intro UN_I[of "(A0, B0)"]) auto 
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qed auto 
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qed (metis A B topological_basis_open open_Times) 
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53255  180 

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subsection {* Countable Basis *} 
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locale countable_basis = 
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fixes B :: "'a::topological_space set set" 
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assumes is_basis: "topological_basis B" 
53282  186 
and countable_basis: "countable B" 
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begin 
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lemma open_countable_basis_ex: 
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assumes "open X" 
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shows "\<exists>B' \<subseteq> B. X = Union B'" 
53255  192 
using assms countable_basis is_basis 
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unfolding topological_basis_def by blast 

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lemma open_countable_basisE: 
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assumes "open X" 
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obtains B' where "B' \<subseteq> B" "X = Union B'" 
53255  198 
using assms open_countable_basis_ex 
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by (atomize_elim) simp 

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lemma countable_dense_exists: 
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"\<exists>D::'a set. countable D \<and> (\<forall>X. open X \<longrightarrow> X \<noteq> {} \<longrightarrow> (\<exists>d \<in> D. d \<in> X))" 
50087  203 
proof  
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let ?f = "(\<lambda>B'. SOME x. x \<in> B')" 
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have "countable (?f ` B)" using countable_basis by simp 
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with basis_dense[OF is_basis, of ?f] show ?thesis 
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by (intro exI[where x="?f ` B"]) (metis (mono_tags) all_not_in_conv imageI someI) 
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qed 
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lemma countable_dense_setE: 

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obtains D :: "'a set" 
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where "countable D" "\<And>X. open X \<Longrightarrow> X \<noteq> {} \<Longrightarrow> \<exists>d \<in> D. d \<in> X" 
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using countable_dense_exists by blast 
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50087  215 
end 
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50883  217 
lemma (in first_countable_topology) first_countable_basisE: 
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obtains A where "countable A" "\<And>a. a \<in> A \<Longrightarrow> x \<in> a" "\<And>a. a \<in> A \<Longrightarrow> open a" 

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"\<And>S. open S \<Longrightarrow> x \<in> S \<Longrightarrow> (\<exists>a\<in>A. a \<subseteq> S)" 

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using first_countable_basis[of x] 

51473  221 
apply atomize_elim 
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apply (elim exE) 

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apply (rule_tac x="range A" in exI) 

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apply auto 

225 
done 

50883  226 

51105  227 
lemma (in first_countable_topology) first_countable_basis_Int_stableE: 
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obtains A where "countable A" "\<And>a. a \<in> A \<Longrightarrow> x \<in> a" "\<And>a. a \<in> A \<Longrightarrow> open a" 

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"\<And>S. open S \<Longrightarrow> x \<in> S \<Longrightarrow> (\<exists>a\<in>A. a \<subseteq> S)" 

230 
"\<And>a b. a \<in> A \<Longrightarrow> b \<in> A \<Longrightarrow> a \<inter> b \<in> A" 

231 
proof atomize_elim 

55522  232 
obtain A' where A': 
233 
"countable A'" 

234 
"\<And>a. a \<in> A' \<Longrightarrow> x \<in> a" 

235 
"\<And>a. a \<in> A' \<Longrightarrow> open a" 

236 
"\<And>S. open S \<Longrightarrow> x \<in> S \<Longrightarrow> \<exists>a\<in>A'. a \<subseteq> S" 

237 
by (rule first_countable_basisE) blast 

51105  238 
def A \<equiv> "(\<lambda>N. \<Inter>((\<lambda>n. from_nat_into A' n) ` N)) ` (Collect finite::nat set set)" 
53255  239 
then show "\<exists>A. countable A \<and> (\<forall>a. a \<in> A \<longrightarrow> x \<in> a) \<and> (\<forall>a. a \<in> A \<longrightarrow> open a) \<and> 
51105  240 
(\<forall>S. open S \<longrightarrow> x \<in> S \<longrightarrow> (\<exists>a\<in>A. a \<subseteq> S)) \<and> (\<forall>a b. a \<in> A \<longrightarrow> b \<in> A \<longrightarrow> a \<inter> b \<in> A)" 
241 
proof (safe intro!: exI[where x=A]) 

53255  242 
show "countable A" 
243 
unfolding A_def by (intro countable_image countable_Collect_finite) 

244 
fix a 

245 
assume "a \<in> A" 

246 
then show "x \<in> a" "open a" 

247 
using A'(4)[OF open_UNIV] by (auto simp: A_def intro: A' from_nat_into) 

51105  248 
next 
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249 
let ?int = "\<lambda>N. \<Inter>(from_nat_into A' ` N)" 
53255  250 
fix a b 
251 
assume "a \<in> A" "b \<in> A" 

252 
then obtain N M where "a = ?int N" "b = ?int M" "finite (N \<union> M)" 

253 
by (auto simp: A_def) 

254 
then show "a \<inter> b \<in> A" 

255 
by (auto simp: A_def intro!: image_eqI[where x="N \<union> M"]) 

51105  256 
next 
53255  257 
fix S 
258 
assume "open S" "x \<in> S" 

259 
then obtain a where a: "a\<in>A'" "a \<subseteq> S" using A' by blast 

260 
then show "\<exists>a\<in>A. a \<subseteq> S" using a A' 

51105  261 
by (intro bexI[where x=a]) (auto simp: A_def intro: image_eqI[where x="{to_nat_on A' a}"]) 
262 
qed 

263 
qed 

264 

51473  265 
lemma (in topological_space) first_countableI: 
53255  266 
assumes "countable A" 
267 
and 1: "\<And>a. a \<in> A \<Longrightarrow> x \<in> a" "\<And>a. a \<in> A \<Longrightarrow> open a" 

268 
and 2: "\<And>S. open S \<Longrightarrow> x \<in> S \<Longrightarrow> \<exists>a\<in>A. a \<subseteq> S" 

51473  269 
shows "\<exists>A::nat \<Rightarrow> 'a set. (\<forall>i. x \<in> A i \<and> open (A i)) \<and> (\<forall>S. open S \<and> x \<in> S \<longrightarrow> (\<exists>i. A i \<subseteq> S))" 
270 
proof (safe intro!: exI[of _ "from_nat_into A"]) 

53255  271 
fix i 
51473  272 
have "A \<noteq> {}" using 2[of UNIV] by auto 
53255  273 
show "x \<in> from_nat_into A i" "open (from_nat_into A i)" 
274 
using range_from_nat_into_subset[OF `A \<noteq> {}`] 1 by auto 

275 
next 

276 
fix S 

277 
assume "open S" "x\<in>S" from 2[OF this] 

278 
show "\<exists>i. from_nat_into A i \<subseteq> S" 

279 
using subset_range_from_nat_into[OF `countable A`] by auto 

51473  280 
qed 
51350  281 

50883  282 
instance prod :: (first_countable_topology, first_countable_topology) first_countable_topology 
283 
proof 

284 
fix x :: "'a \<times> 'b" 

55522  285 
obtain A where A: 
286 
"countable A" 

287 
"\<And>a. a \<in> A \<Longrightarrow> fst x \<in> a" 

288 
"\<And>a. a \<in> A \<Longrightarrow> open a" 

289 
"\<And>S. open S \<Longrightarrow> fst x \<in> S \<Longrightarrow> \<exists>a\<in>A. a \<subseteq> S" 

290 
by (rule first_countable_basisE[of "fst x"]) blast 

291 
obtain B where B: 

292 
"countable B" 

293 
"\<And>a. a \<in> B \<Longrightarrow> snd x \<in> a" 

294 
"\<And>a. a \<in> B \<Longrightarrow> open a" 

295 
"\<And>S. open S \<Longrightarrow> snd x \<in> S \<Longrightarrow> \<exists>a\<in>B. a \<subseteq> S" 

296 
by (rule first_countable_basisE[of "snd x"]) blast 

53282  297 
show "\<exists>A::nat \<Rightarrow> ('a \<times> 'b) set. 
298 
(\<forall>i. x \<in> A i \<and> open (A i)) \<and> (\<forall>S. open S \<and> x \<in> S \<longrightarrow> (\<exists>i. A i \<subseteq> S))" 

51473  299 
proof (rule first_countableI[of "(\<lambda>(a, b). a \<times> b) ` (A \<times> B)"], safe) 
53255  300 
fix a b 
301 
assume x: "a \<in> A" "b \<in> B" 

53640  302 
with A(2, 3)[of a] B(2, 3)[of b] show "x \<in> a \<times> b" and "open (a \<times> b)" 
303 
unfolding mem_Times_iff 

304 
by (auto intro: open_Times) 

50883  305 
next 
53255  306 
fix S 
307 
assume "open S" "x \<in> S" 

55522  308 
then obtain a' b' where a'b': "open a'" "open b'" "x \<in> a' \<times> b'" "a' \<times> b' \<subseteq> S" 
309 
by (rule open_prod_elim) 

310 
moreover 

311 
from a'b' A(4)[of a'] B(4)[of b'] 

312 
obtain a b where "a \<in> A" "a \<subseteq> a'" "b \<in> B" "b \<subseteq> b'" 

313 
by auto 

314 
ultimately 

315 
show "\<exists>a\<in>(\<lambda>(a, b). a \<times> b) ` (A \<times> B). a \<subseteq> S" 

50883  316 
by (auto intro!: bexI[of _ "a \<times> b"] bexI[of _ a] bexI[of _ b]) 
317 
qed (simp add: A B) 

318 
qed 

319 

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320 
class second_countable_topology = topological_space + 
53282  321 
assumes ex_countable_subbasis: 
322 
"\<exists>B::'a::topological_space set set. countable B \<and> open = generate_topology B" 

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323 
begin 
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324 

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325 
lemma ex_countable_basis: "\<exists>B::'a set set. countable B \<and> topological_basis B" 
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326 
proof  
53255  327 
from ex_countable_subbasis obtain B where B: "countable B" "open = generate_topology B" 
328 
by blast 

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329 
let ?B = "Inter ` {b. finite b \<and> b \<subseteq> B }" 
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330 

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331 
show ?thesis 
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332 
proof (intro exI conjI) 
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333 
show "countable ?B" 
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334 
by (intro countable_image countable_Collect_finite_subset B) 
53255  335 
{ 
336 
fix S 

337 
assume "open S" 

51343
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338 
then have "\<exists>B'\<subseteq>{b. finite b \<and> b \<subseteq> B}. (\<Union>b\<in>B'. \<Inter>b) = S" 
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339 
unfolding B 
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340 
proof induct 
53255  341 
case UNIV 
342 
show ?case by (intro exI[of _ "{{}}"]) simp 

51343
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343 
next 
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344 
case (Int a b) 
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345 
then obtain x y where x: "a = UNION x Inter" "\<And>i. i \<in> x \<Longrightarrow> finite i \<and> i \<subseteq> B" 
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346 
and y: "b = UNION y Inter" "\<And>i. i \<in> y \<Longrightarrow> finite i \<and> i \<subseteq> B" 
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347 
by blast 
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348 
show ?case 
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349 
unfolding x y Int_UN_distrib2 
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350 
by (intro exI[of _ "{i \<union> j i j. i \<in> x \<and> j \<in> y}"]) (auto dest: x(2) y(2)) 
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351 
next 
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352 
case (UN K) 
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353 
then have "\<forall>k\<in>K. \<exists>B'\<subseteq>{b. finite b \<and> b \<subseteq> B}. UNION B' Inter = k" by auto 
55522  354 
then obtain k where 
355 
"\<forall>ka\<in>K. k ka \<subseteq> {b. finite b \<and> b \<subseteq> B} \<and> UNION (k ka) Inter = ka" 

356 
unfolding bchoice_iff .. 

51343
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357 
then show "\<exists>B'\<subseteq>{b. finite b \<and> b \<subseteq> B}. UNION B' Inter = \<Union>K" 
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358 
by (intro exI[of _ "UNION K k"]) auto 
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359 
next 
53255  360 
case (Basis S) 
361 
then show ?case 

51343
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362 
by (intro exI[of _ "{{S}}"]) auto 
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363 
qed 
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364 
then have "(\<exists>B'\<subseteq>Inter ` {b. finite b \<and> b \<subseteq> B}. \<Union>B' = S)" 
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365 
unfolding subset_image_iff by blast } 
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366 
then show "topological_basis ?B" 
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367 
unfolding topological_space_class.topological_basis_def 
53282  368 
by (safe intro!: topological_space_class.open_Inter) 
51343
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369 
(simp_all add: B generate_topology.Basis subset_eq) 
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370 
qed 
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371 
qed 
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372 

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373 
end 
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374 

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375 
sublocale second_countable_topology < 
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376 
countable_basis "SOME B. countable B \<and> topological_basis B" 
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377 
using someI_ex[OF ex_countable_basis] 
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378 
by unfold_locales safe 
50094
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379 

50882
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380 
instance prod :: (second_countable_topology, second_countable_topology) second_countable_topology 
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381 
proof 
a382bf90867e
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382 
obtain A :: "'a set set" where "countable A" "topological_basis A" 
a382bf90867e
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383 
using ex_countable_basis by auto 
a382bf90867e
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384 
moreover 
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385 
obtain B :: "'b set set" where "countable B" "topological_basis B" 
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386 
using ex_countable_basis by auto 
51343
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387 
ultimately show "\<exists>B::('a \<times> 'b) set set. countable B \<and> open = generate_topology B" 
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388 
by (auto intro!: exI[of _ "(\<lambda>(a, b). a \<times> b) ` (A \<times> B)"] topological_basis_prod 
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389 
topological_basis_imp_subbasis) 
50882
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390 
qed 
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391 

50883  392 
instance second_countable_topology \<subseteq> first_countable_topology 
393 
proof 

394 
fix x :: 'a 

395 
def B \<equiv> "SOME B::'a set set. countable B \<and> topological_basis B" 

396 
then have B: "countable B" "topological_basis B" 

397 
using countable_basis is_basis 

398 
by (auto simp: countable_basis is_basis) 

53282  399 
then show "\<exists>A::nat \<Rightarrow> 'a set. 
400 
(\<forall>i. x \<in> A i \<and> open (A i)) \<and> (\<forall>S. open S \<and> x \<in> S \<longrightarrow> (\<exists>i. A i \<subseteq> S))" 

51473  401 
by (intro first_countableI[of "{b\<in>B. x \<in> b}"]) 
402 
(fastforce simp: topological_space_class.topological_basis_def)+ 

50883  403 
qed 
404 

53255  405 

50087  406 
subsection {* Polish spaces *} 
407 

408 
text {* Textbooks define Polish spaces as completely metrizable. 

409 
We assume the topology to be complete for a given metric. *} 

410 

50881
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411 
class polish_space = complete_space + second_countable_topology 
50087  412 

44517  413 
subsection {* General notion of a topology as a value *} 
33175  414 

53255  415 
definition "istopology L \<longleftrightarrow> 
416 
L {} \<and> (\<forall>S T. L S \<longrightarrow> L T \<longrightarrow> L (S \<inter> T)) \<and> (\<forall>K. Ball K L \<longrightarrow> L (\<Union> K))" 

417 

49834  418 
typedef 'a topology = "{L::('a set) \<Rightarrow> bool. istopology L}" 
33175  419 
morphisms "openin" "topology" 
420 
unfolding istopology_def by blast 

421 

422 
lemma istopology_open_in[intro]: "istopology(openin U)" 

423 
using openin[of U] by blast 

424 

425 
lemma topology_inverse': "istopology U \<Longrightarrow> openin (topology U) = U" 

44170
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426 
using topology_inverse[unfolded mem_Collect_eq] . 
33175  427 

428 
lemma topology_inverse_iff: "istopology U \<longleftrightarrow> openin (topology U) = U" 

429 
using topology_inverse[of U] istopology_open_in[of "topology U"] by auto 

430 

431 
lemma topology_eq: "T1 = T2 \<longleftrightarrow> (\<forall>S. openin T1 S \<longleftrightarrow> openin T2 S)" 

53255  432 
proof 
433 
assume "T1 = T2" 

434 
then show "\<forall>S. openin T1 S \<longleftrightarrow> openin T2 S" by simp 

435 
next 

436 
assume H: "\<forall>S. openin T1 S \<longleftrightarrow> openin T2 S" 

437 
then have "openin T1 = openin T2" by (simp add: fun_eq_iff) 

438 
then have "topology (openin T1) = topology (openin T2)" by simp 

439 
then show "T1 = T2" unfolding openin_inverse . 

33175  440 
qed 
441 

442 
text{* Infer the "universe" from union of all sets in the topology. *} 

443 

53640  444 
definition "topspace T = \<Union>{S. openin T S}" 
33175  445 

44210
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446 
subsubsection {* Main properties of open sets *} 
33175  447 

448 
lemma openin_clauses: 

449 
fixes U :: "'a topology" 

53282  450 
shows 
451 
"openin U {}" 

452 
"\<And>S T. openin U S \<Longrightarrow> openin U T \<Longrightarrow> openin U (S\<inter>T)" 

453 
"\<And>K. (\<forall>S \<in> K. openin U S) \<Longrightarrow> openin U (\<Union>K)" 

454 
using openin[of U] unfolding istopology_def mem_Collect_eq by fast+ 

33175  455 

456 
lemma openin_subset[intro]: "openin U S \<Longrightarrow> S \<subseteq> topspace U" 

457 
unfolding topspace_def by blast 

53255  458 

459 
lemma openin_empty[simp]: "openin U {}" 

460 
by (simp add: openin_clauses) 

33175  461 

462 
lemma openin_Int[intro]: "openin U S \<Longrightarrow> openin U T \<Longrightarrow> openin U (S \<inter> T)" 

36362
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463 
using openin_clauses by simp 
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464 

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465 
lemma openin_Union[intro]: "(\<forall>S \<in>K. openin U S) \<Longrightarrow> openin U (\<Union> K)" 
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466 
using openin_clauses by simp 
33175  467 

468 
lemma openin_Un[intro]: "openin U S \<Longrightarrow> openin U T \<Longrightarrow> openin U (S \<union> T)" 

469 
using openin_Union[of "{S,T}" U] by auto 

470 

53255  471 
lemma openin_topspace[intro, simp]: "openin U (topspace U)" 
472 
by (simp add: openin_Union topspace_def) 

33175  473 

49711  474 
lemma openin_subopen: "openin U S \<longleftrightarrow> (\<forall>x \<in> S. \<exists>T. openin U T \<and> x \<in> T \<and> T \<subseteq> S)" 
475 
(is "?lhs \<longleftrightarrow> ?rhs") 

36584  476 
proof 
49711  477 
assume ?lhs 
478 
then show ?rhs by auto 

36584  479 
next 
480 
assume H: ?rhs 

481 
let ?t = "\<Union>{T. openin U T \<and> T \<subseteq> S}" 

482 
have "openin U ?t" by (simp add: openin_Union) 

483 
also have "?t = S" using H by auto 

484 
finally show "openin U S" . 

33175  485 
qed 
486 

49711  487 

44210
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huffman
parents:
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diff
changeset

488 
subsubsection {* Closed sets *} 
33175  489 

490 
definition "closedin U S \<longleftrightarrow> S \<subseteq> topspace U \<and> openin U (topspace U  S)" 

491 

53255  492 
lemma closedin_subset: "closedin U S \<Longrightarrow> S \<subseteq> topspace U" 
493 
by (metis closedin_def) 

494 

495 
lemma closedin_empty[simp]: "closedin U {}" 

496 
by (simp add: closedin_def) 

497 

498 
lemma closedin_topspace[intro, simp]: "closedin U (topspace U)" 

499 
by (simp add: closedin_def) 

500 

33175  501 
lemma closedin_Un[intro]: "closedin U S \<Longrightarrow> closedin U T \<Longrightarrow> closedin U (S \<union> T)" 
502 
by (auto simp add: Diff_Un closedin_def) 

503 

53255  504 
lemma Diff_Inter[intro]: "A  \<Inter>S = \<Union> {A  ss. s\<in>S}" 
505 
by auto 

506 

507 
lemma closedin_Inter[intro]: 

508 
assumes Ke: "K \<noteq> {}" 

509 
and Kc: "\<forall>S \<in>K. closedin U S" 

510 
shows "closedin U (\<Inter> K)" 

511 
using Ke Kc unfolding closedin_def Diff_Inter by auto 

33175  512 

513 
lemma closedin_Int[intro]: "closedin U S \<Longrightarrow> closedin U T \<Longrightarrow> closedin U (S \<inter> T)" 

514 
using closedin_Inter[of "{S,T}" U] by auto 

515 

53255  516 
lemma Diff_Diff_Int: "A  (A  B) = A \<inter> B" 
517 
by blast 

518 

33175  519 
lemma openin_closedin_eq: "openin U S \<longleftrightarrow> S \<subseteq> topspace U \<and> closedin U (topspace U  S)" 
520 
apply (auto simp add: closedin_def Diff_Diff_Int inf_absorb2) 

521 
apply (metis openin_subset subset_eq) 

522 
done 

523 

53255  524 
lemma openin_closedin: "S \<subseteq> topspace U \<Longrightarrow> (openin U S \<longleftrightarrow> closedin U (topspace U  S))" 
33175  525 
by (simp add: openin_closedin_eq) 
526 

53255  527 
lemma openin_diff[intro]: 
528 
assumes oS: "openin U S" 

529 
and cT: "closedin U T" 

530 
shows "openin U (S  T)" 

531 
proof  

33175  532 
have "S  T = S \<inter> (topspace U  T)" using openin_subset[of U S] oS cT 
533 
by (auto simp add: topspace_def openin_subset) 

53282  534 
then show ?thesis using oS cT 
535 
by (auto simp add: closedin_def) 

33175  536 
qed 
537 

53255  538 
lemma closedin_diff[intro]: 
539 
assumes oS: "closedin U S" 

540 
and cT: "openin U T" 

541 
shows "closedin U (S  T)" 

542 
proof  

543 
have "S  T = S \<inter> (topspace U  T)" 

53282  544 
using closedin_subset[of U S] oS cT by (auto simp add: topspace_def) 
53255  545 
then show ?thesis 
546 
using oS cT by (auto simp add: openin_closedin_eq) 

547 
qed 

548 

33175  549 

44210
eba74571833b
Topology_Euclidean_Space.thy: organize section headings
huffman
parents:
44207
diff
changeset

550 
subsubsection {* Subspace topology *} 
44170
510ac30f44c0
make Multivariate_Analysis work with separate set type
huffman
parents:
44167
diff
changeset

551 

510ac30f44c0
make Multivariate_Analysis work with separate set type
huffman
parents:
44167
diff
changeset

552 
definition "subtopology U V = topology (\<lambda>T. \<exists>S. T = S \<inter> V \<and> openin U S)" 
510ac30f44c0
make Multivariate_Analysis work with separate set type
huffman
parents:
44167
diff
changeset

553 

510ac30f44c0
make Multivariate_Analysis work with separate set type
huffman
parents:
44167
diff
changeset

554 
lemma istopology_subtopology: "istopology (\<lambda>T. \<exists>S. T = S \<inter> V \<and> openin U S)" 
510ac30f44c0
make Multivariate_Analysis work with separate set type
huffman
parents:
44167
diff
changeset

555 
(is "istopology ?L") 
53255  556 
proof  
44170
510ac30f44c0
make Multivariate_Analysis work with separate set type
huffman
parents:
44167
diff
changeset

557 
have "?L {}" by blast 
53255  558 
{ 
559 
fix A B 

560 
assume A: "?L A" and B: "?L B" 

561 
from A B obtain Sa and Sb where Sa: "openin U Sa" "A = Sa \<inter> V" and Sb: "openin U Sb" "B = Sb \<inter> V" 

562 
by blast 

563 
have "A \<inter> B = (Sa \<inter> Sb) \<inter> V" "openin U (Sa \<inter> Sb)" 

564 
using Sa Sb by blast+ 

565 
then have "?L (A \<inter> B)" by blast 

566 
} 

33175  567 
moreover 
53255  568 
{ 
53282  569 
fix K 
570 
assume K: "K \<subseteq> Collect ?L" 

44170
510ac30f44c0
make Multivariate_Analysis work with separate set type
huffman
parents:
44167
diff
changeset

571 
have th0: "Collect ?L = (\<lambda>S. S \<inter> V) ` Collect (openin U)" 
55775  572 
by blast 
33175  573 
from K[unfolded th0 subset_image_iff] 
53255  574 
obtain Sk where Sk: "Sk \<subseteq> Collect (openin U)" "K = (\<lambda>S. S \<inter> V) ` Sk" 
575 
by blast 

576 
have "\<Union>K = (\<Union>Sk) \<inter> V" 

577 
using Sk by auto 

578 
moreover have "openin U (\<Union> Sk)" 

579 
using Sk by (auto simp add: subset_eq) 

580 
ultimately have "?L (\<Union>K)" by blast 

581 
} 

44170
510ac30f44c0
make Multivariate_Analysis work with separate set type
huffman
parents:
44167
diff
changeset

582 
ultimately show ?thesis 
510ac30f44c0
make Multivariate_Analysis work with separate set type
huffman
parents:
44167
diff
changeset

583 
unfolding subset_eq mem_Collect_eq istopology_def by blast 
33175  584 
qed 
585 

53255  586 
lemma openin_subtopology: "openin (subtopology U V) S \<longleftrightarrow> (\<exists>T. openin U T \<and> S = T \<inter> V)" 
33175  587 
unfolding subtopology_def topology_inverse'[OF istopology_subtopology] 
44170
510ac30f44c0
make Multivariate_Analysis work with separate set type
huffman
parents:
44167
diff
changeset

588 
by auto 
33175  589 

53255  590 
lemma topspace_subtopology: "topspace (subtopology U V) = topspace U \<inter> V" 
33175  591 
by (auto simp add: topspace_def openin_subtopology) 
592 

53255  593 
lemma closedin_subtopology: "closedin (subtopology U V) S \<longleftrightarrow> (\<exists>T. closedin U T \<and> S = T \<inter> V)" 
33175  594 
unfolding closedin_def topspace_subtopology 
55775  595 
by (auto simp add: openin_subtopology) 
33175  596 

597 
lemma openin_subtopology_refl: "openin (subtopology U V) V \<longleftrightarrow> V \<subseteq> topspace U" 

598 
unfolding openin_subtopology 

55775  599 
by auto (metis IntD1 in_mono openin_subset) 
49711  600 

601 
lemma subtopology_superset: 

602 
assumes UV: "topspace U \<subseteq> V" 

33175  603 
shows "subtopology U V = U" 
53255  604 
proof  
605 
{ 

606 
fix S 

607 
{ 

608 
fix T 

609 
assume T: "openin U T" "S = T \<inter> V" 

610 
from T openin_subset[OF T(1)] UV have eq: "S = T" 

611 
by blast 

612 
have "openin U S" 

613 
unfolding eq using T by blast 

614 
} 

33175  615 
moreover 
53255  616 
{ 
617 
assume S: "openin U S" 

618 
then have "\<exists>T. openin U T \<and> S = T \<inter> V" 

619 
using openin_subset[OF S] UV by auto 

620 
} 

621 
ultimately have "(\<exists>T. openin U T \<and> S = T \<inter> V) \<longleftrightarrow> openin U S" 

622 
by blast 

623 
} 

624 
then show ?thesis 

625 
unfolding topology_eq openin_subtopology by blast 

33175  626 
qed 
627 

628 
lemma subtopology_topspace[simp]: "subtopology U (topspace U) = U" 

629 
by (simp add: subtopology_superset) 

630 

631 
lemma subtopology_UNIV[simp]: "subtopology U UNIV = U" 

632 
by (simp add: subtopology_superset) 

633 

53255  634 

44210
eba74571833b
Topology_Euclidean_Space.thy: organize section headings
huffman
parents:
44207
diff
changeset

635 
subsubsection {* The standard Euclidean topology *} 
33175  636 

53255  637 
definition euclidean :: "'a::topological_space topology" 
638 
where "euclidean = topology open" 

33175  639 

640 
lemma open_openin: "open S \<longleftrightarrow> openin euclidean S" 

641 
unfolding euclidean_def 

642 
apply (rule cong[where x=S and y=S]) 

643 
apply (rule topology_inverse[symmetric]) 

644 
apply (auto simp add: istopology_def) 

44170
510ac30f44c0
make Multivariate_Analysis work with separate set type
huffman
parents:
44167
diff
changeset

645 
done 
33175  646 

647 
lemma topspace_euclidean: "topspace euclidean = UNIV" 

648 
apply (simp add: topspace_def) 

39302
d7728f65b353
renamed lemmas: ext_iff > fun_eq_iff, set_ext_iff > set_eq_iff, set_ext > set_eqI
nipkow
parents:
39198
diff
changeset

649 
apply (rule set_eqI) 
53255  650 
apply (auto simp add: open_openin[symmetric]) 
651 
done 

33175  652 

653 
lemma topspace_euclidean_subtopology[simp]: "topspace (subtopology euclidean S) = S" 

654 
by (simp add: topspace_euclidean topspace_subtopology) 

655 

656 
lemma closed_closedin: "closed S \<longleftrightarrow> closedin euclidean S" 

657 
by (simp add: closed_def closedin_def topspace_euclidean open_openin Compl_eq_Diff_UNIV) 

658 

659 
lemma open_subopen: "open S \<longleftrightarrow> (\<forall>x\<in>S. \<exists>T. open T \<and> x \<in> T \<and> T \<subseteq> S)" 

660 
by (simp add: open_openin openin_subopen[symmetric]) 

661 

44210
eba74571833b
Topology_Euclidean_Space.thy: organize section headings
huffman
parents:
44207
diff
changeset

662 
text {* Basic "localization" results are handy for connectedness. *} 
eba74571833b
Topology_Euclidean_Space.thy: organize section headings
huffman
parents:
44207
diff
changeset

663 

eba74571833b
Topology_Euclidean_Space.thy: organize section headings
huffman
parents:
44207
diff
changeset

664 
lemma openin_open: "openin (subtopology euclidean U) S \<longleftrightarrow> (\<exists>T. open T \<and> (S = U \<inter> T))" 
eba74571833b
Topology_Euclidean_Space.thy: organize section headings
huffman
parents:
44207
diff
changeset

665 
by (auto simp add: openin_subtopology open_openin[symmetric]) 
eba74571833b
Topology_Euclidean_Space.thy: organize section headings
huffman
parents:
44207
diff
changeset

666 

eba74571833b
Topology_Euclidean_Space.thy: organize section headings
huffman
parents:
44207
diff
changeset

667 
lemma openin_open_Int[intro]: "open S \<Longrightarrow> openin (subtopology euclidean U) (U \<inter> S)" 
eba74571833b
Topology_Euclidean_Space.thy: organize section headings
huffman
parents:
44207
diff
changeset

668 
by (auto simp add: openin_open) 
eba74571833b
Topology_Euclidean_Space.thy: organize section headings
huffman
parents:
44207
diff
changeset

669 

eba74571833b
Topology_Euclidean_Space.thy: organize section headings
huffman
parents:
44207
diff
changeset

670 
lemma open_openin_trans[trans]: 
53255  671 
"open S \<Longrightarrow> open T \<Longrightarrow> T \<subseteq> S \<Longrightarrow> openin (subtopology euclidean S) T" 
44210
eba74571833b
Topology_Euclidean_Space.thy: organize section headings
huffman
parents:
44207
diff
changeset

672 
by (metis Int_absorb1 openin_open_Int) 
eba74571833b
Topology_Euclidean_Space.thy: organize section headings
huffman
parents:
44207
diff
changeset

673 

53255  674 
lemma open_subset: "S \<subseteq> T \<Longrightarrow> open S \<Longrightarrow> openin (subtopology euclidean T) S" 
44210
eba74571833b
Topology_Euclidean_Space.thy: organize section headings
huffman
parents:
44207
diff
changeset

675 
by (auto simp add: openin_open) 
eba74571833b
Topology_Euclidean_Space.thy: organize section headings
huffman
parents:
44207
diff
changeset

676 

eba74571833b
Topology_Euclidean_Space.thy: organize section headings
huffman
parents:
44207
diff
changeset

677 
lemma closedin_closed: "closedin (subtopology euclidean U) S \<longleftrightarrow> (\<exists>T. closed T \<and> S = U \<inter> T)" 
eba74571833b
Topology_Euclidean_Space.thy: organize section headings
huffman
parents:
44207
diff
changeset

678 
by (simp add: closedin_subtopology closed_closedin Int_ac) 
eba74571833b
Topology_Euclidean_Space.thy: organize section headings
huffman
parents:
44207
diff
changeset

679 

53291  680 
lemma closedin_closed_Int: "closed S \<Longrightarrow> closedin (subtopology euclidean U) (U \<inter> S)" 
44210
eba74571833b
Topology_Euclidean_Space.thy: organize section headings
huffman
parents:
44207
diff
changeset

681 
by (metis closedin_closed) 
eba74571833b
Topology_Euclidean_Space.thy: organize section headings
huffman
parents:
44207
diff
changeset

682 

53282  683 
lemma closed_closedin_trans: 
684 
"closed S \<Longrightarrow> closed T \<Longrightarrow> T \<subseteq> S \<Longrightarrow> closedin (subtopology euclidean S) T" 

55775  685 
by (metis closedin_closed inf.absorb2) 
44210
eba74571833b
Topology_Euclidean_Space.thy: organize section headings
huffman
parents:
44207
diff
changeset

686 

eba74571833b
Topology_Euclidean_Space.thy: organize section headings
huffman
parents:
44207
diff
changeset

687 
lemma closed_subset: "S \<subseteq> T \<Longrightarrow> closed S \<Longrightarrow> closedin (subtopology euclidean T) S" 
eba74571833b
Topology_Euclidean_Space.thy: organize section headings
huffman
parents:
44207
diff
changeset

688 
by (auto simp add: closedin_closed) 
eba74571833b
Topology_Euclidean_Space.thy: organize section headings
huffman
parents:
44207
diff
changeset

689 

eba74571833b
Topology_Euclidean_Space.thy: organize section headings
huffman
parents:
44207
diff
changeset

690 
lemma openin_euclidean_subtopology_iff: 
eba74571833b
Topology_Euclidean_Space.thy: organize section headings
huffman
parents:
44207
diff
changeset

691 
fixes S U :: "'a::metric_space set" 
53255  692 
shows "openin (subtopology euclidean U) S \<longleftrightarrow> 
693 
S \<subseteq> U \<and> (\<forall>x\<in>S. \<exists>e>0. \<forall>x'\<in>U. dist x' x < e \<longrightarrow> x'\<in> S)" 

694 
(is "?lhs \<longleftrightarrow> ?rhs") 

44210
eba74571833b
Topology_Euclidean_Space.thy: organize section headings
huffman
parents:
44207
diff
changeset

695 
proof 
53255  696 
assume ?lhs 
53282  697 
then show ?rhs 
698 
unfolding openin_open open_dist by blast 

44210
eba74571833b
Topology_Euclidean_Space.thy: organize section headings
huffman
parents:
44207
diff
changeset

699 
next 
eba74571833b
Topology_Euclidean_Space.thy: organize section headings
huffman
parents:
44207
diff
changeset

700 
def T \<equiv> "{x. \<exists>a\<in>S. \<exists>d>0. (\<forall>y\<in>U. dist y a < d \<longrightarrow> y \<in> S) \<and> dist x a < d}" 
eba74571833b
Topology_Euclidean_Space.thy: organize section headings
huffman
parents:
44207
diff
changeset

701 
have 1: "\<forall>x\<in>T. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> T" 
eba74571833b
Topology_Euclidean_Space.thy: organize section headings
huffman
parents:
44207
diff
changeset

702 
unfolding T_def 
eba74571833b
Topology_Euclidean_Space.thy: organize section headings
huffman
parents:
44207
diff
changeset

703 
apply clarsimp 
eba74571833b
Topology_Euclidean_Space.thy: organize section headings
huffman
parents:
44207
diff
changeset

704 
apply (rule_tac x="d  dist x a" in exI) 
eba74571833b
Topology_Euclidean_Space.thy: organize section headings
huffman
parents:
44207
diff
changeset

705 
apply (clarsimp simp add: less_diff_eq) 
55775  706 
by (metis dist_commute dist_triangle_lt) 
53282  707 
assume ?rhs then have 2: "S = U \<inter> T" 
55775  708 
unfolding T_def 
709 
by auto (metis dist_self) 

44210
eba74571833b
Topology_Euclidean_Space.thy: organize section headings
huffman
parents:
44207
diff
changeset

710 
from 1 2 show ?lhs 
eba74571833b
Topology_Euclidean_Space.thy: organize section headings
huffman
parents:
44207
diff
changeset

711 
unfolding openin_open open_dist by fast 
eba74571833b
Topology_Euclidean_Space.thy: organize section headings
huffman
parents:
44207
diff
changeset

712 
qed 
eba74571833b
Topology_Euclidean_Space.thy: organize section headings
huffman
parents:
44207
diff
changeset

713 

eba74571833b
Topology_Euclidean_Space.thy: organize section headings
huffman
parents:
44207
diff
changeset

714 
text {* These "transitivity" results are handy too *} 
eba74571833b
Topology_Euclidean_Space.thy: organize section headings
huffman
parents:
44207
diff
changeset

715 

53255  716 
lemma openin_trans[trans]: 
717 
"openin (subtopology euclidean T) S \<Longrightarrow> openin (subtopology euclidean U) T \<Longrightarrow> 

718 
openin (subtopology euclidean U) S" 

44210
eba74571833b
Topology_Euclidean_Space.thy: organize section headings
huffman
parents:
44207
diff
changeset

719 
unfolding open_openin openin_open by blast 
eba74571833b
Topology_Euclidean_Space.thy: organize section headings
huffman
parents:
44207
diff
changeset

720 

eba74571833b
Topology_Euclidean_Space.thy: organize section headings
huffman
parents:
44207
diff
changeset

721 
lemma openin_open_trans: "openin (subtopology euclidean T) S \<Longrightarrow> open T \<Longrightarrow> open S" 
eba74571833b
Topology_Euclidean_Space.thy: organize section headings
huffman
parents:
44207
diff
changeset

722 
by (auto simp add: openin_open intro: openin_trans) 
eba74571833b
Topology_Euclidean_Space.thy: organize section headings
huffman
parents:
44207
diff
changeset

723 

eba74571833b
Topology_Euclidean_Space.thy: organize section headings
huffman
parents:
44207
diff
changeset

724 
lemma closedin_trans[trans]: 
53255  725 
"closedin (subtopology euclidean T) S \<Longrightarrow> closedin (subtopology euclidean U) T \<Longrightarrow> 
726 
closedin (subtopology euclidean U) S" 

44210
eba74571833b
Topology_Euclidean_Space.thy: organize section headings
huffman
parents:
44207
diff
changeset

727 
by (auto simp add: closedin_closed closed_closedin closed_Inter Int_assoc) 
eba74571833b
Topology_Euclidean_Space.thy: organize section headings
huffman
parents:
44207
diff
changeset

728 

eba74571833b
Topology_Euclidean_Space.thy: organize section headings
huffman
parents:
44207
diff
changeset

729 
lemma closedin_closed_trans: "closedin (subtopology euclidean T) S \<Longrightarrow> closed T \<Longrightarrow> closed S" 
eba74571833b
Topology_Euclidean_Space.thy: organize section headings
huffman
parents:
44207
diff
changeset

730 
by (auto simp add: closedin_closed intro: closedin_trans) 
eba74571833b
Topology_Euclidean_Space.thy: organize section headings
huffman
parents:
44207
diff
changeset

731 

eba74571833b
Topology_Euclidean_Space.thy: organize section headings
huffman
parents:
44207
diff
changeset

732 

eba74571833b
Topology_Euclidean_Space.thy: organize section headings
huffman
parents:
44207
diff
changeset

733 
subsection {* Open and closed balls *} 
33175  734 

53255  735 
definition ball :: "'a::metric_space \<Rightarrow> real \<Rightarrow> 'a set" 
736 
where "ball x e = {y. dist x y < e}" 

737 

738 
definition cball :: "'a::metric_space \<Rightarrow> real \<Rightarrow> 'a set" 

739 
where "cball x e = {y. dist x y \<le> e}" 

33175  740 

45776
714100f5fda4
remove mem_(c)ball_0 and centre_in_(c)ball from simpset, as rules mem_(c)ball always match instead
huffman
parents:
45548
diff
changeset

741 
lemma mem_ball [simp]: "y \<in> ball x e \<longleftrightarrow> dist x y < e" 
714100f5fda4
remove mem_(c)ball_0 and centre_in_(c)ball from simpset, as rules mem_(c)ball always match instead
huffman
parents:
45548
diff
changeset

742 
by (simp add: ball_def) 
714100f5fda4
remove mem_(c)ball_0 and centre_in_(c)ball from simpset, as rules mem_(c)ball always match instead
huffman
parents:
45548
diff
changeset

743 

714100f5fda4
remove mem_(c)ball_0 and centre_in_(c)ball from simpset, as rules mem_(c)ball always match instead
huffman
parents:
45548
diff
changeset

744 
lemma mem_cball [simp]: "y \<in> cball x e \<longleftrightarrow> dist x y \<le> e" 
714100f5fda4
remove mem_(c)ball_0 and centre_in_(c)ball from simpset, as rules mem_(c)ball always match instead
huffman
parents:
45548
diff
changeset

745 
by (simp add: cball_def) 
714100f5fda4
remove mem_(c)ball_0 and centre_in_(c)ball from simpset, as rules mem_(c)ball always match instead
huffman
parents:
45548
diff
changeset

746 

714100f5fda4
remove mem_(c)ball_0 and centre_in_(c)ball from simpset, as rules mem_(c)ball always match instead
huffman
parents:
45548
diff
changeset

747 
lemma mem_ball_0: 
33175  748 
fixes x :: "'a::real_normed_vector" 
749 
shows "x \<in> ball 0 e \<longleftrightarrow> norm x < e" 

750 
by (simp add: dist_norm) 

751 

45776
714100f5fda4
remove mem_(c)ball_0 and centre_in_(c)ball from simpset, as rules mem_(c)ball always match instead
huffman
parents:
45548
diff
changeset

752 
lemma mem_cball_0: 
33175  753 
fixes x :: "'a::real_normed_vector" 
754 
shows "x \<in> cball 0 e \<longleftrightarrow> norm x \<le> e" 

755 
by (simp add: dist_norm) 

756 

45776
714100f5fda4
remove mem_(c)ball_0 and centre_in_(c)ball from simpset, as rules mem_(c)ball always match instead
huffman
parents:
45548
diff
changeset

757 
lemma centre_in_ball: "x \<in> ball x e \<longleftrightarrow> 0 < e" 
714100f5fda4
remove mem_(c)ball_0 and centre_in_(c)ball from simpset, as rules mem_(c)ball always match instead
huffman
parents:
45548
diff
changeset

758 
by simp 
714100f5fda4
remove mem_(c)ball_0 and centre_in_(c)ball from simpset, as rules mem_(c)ball always match instead
huffman
parents:
45548
diff
changeset

759 

714100f5fda4
remove mem_(c)ball_0 and centre_in_(c)ball from simpset, as rules mem_(c)ball always match instead
huffman
parents:
45548
diff
changeset

760 
lemma centre_in_cball: "x \<in> cball x e \<longleftrightarrow> 0 \<le> e" 
714100f5fda4
remove mem_(c)ball_0 and centre_in_(c)ball from simpset, as rules mem_(c)ball always match instead
huffman
parents:
45548
diff
changeset

761 
by simp 
714100f5fda4
remove mem_(c)ball_0 and centre_in_(c)ball from simpset, as rules mem_(c)ball always match instead
huffman
parents:
45548
diff
changeset

762 

53255  763 
lemma ball_subset_cball[simp,intro]: "ball x e \<subseteq> cball x e" 
764 
by (simp add: subset_eq) 

765 

53282  766 
lemma subset_ball[intro]: "d \<le> e \<Longrightarrow> ball x d \<subseteq> ball x e" 
53255  767 
by (simp add: subset_eq) 
768 

53282  769 
lemma subset_cball[intro]: "d \<le> e \<Longrightarrow> cball x d \<subseteq> cball x e" 
53255  770 
by (simp add: subset_eq) 
771 

33175  772 
lemma ball_max_Un: "ball a (max r s) = ball a r \<union> ball a s" 
39302
d7728f65b353
renamed lemmas: ext_iff > fun_eq_iff, set_ext_iff > set_eq_iff, set_ext > set_eqI
nipkow
parents:
39198
diff
changeset

773 
by (simp add: set_eq_iff) arith 
33175  774 

775 
lemma ball_min_Int: "ball a (min r s) = ball a r \<inter> ball a s" 

39302
d7728f65b353
renamed lemmas: ext_iff > fun_eq_iff, set_ext_iff > set_eq_iff, set_ext > set_eqI
nipkow
parents:
39198
diff
changeset

776 
by (simp add: set_eq_iff) 
33175  777 

53255  778 
lemma diff_less_iff: 
779 
"(a::real)  b > 0 \<longleftrightarrow> a > b" 

33175  780 
"(a::real)  b < 0 \<longleftrightarrow> a < b" 
53255  781 
"a  b < c \<longleftrightarrow> a < c + b" "a  b > c \<longleftrightarrow> a > c + b" 
782 
by arith+ 

783 

784 
lemma diff_le_iff: 

785 
"(a::real)  b \<ge> 0 \<longleftrightarrow> a \<ge> b" 

786 
"(a::real)  b \<le> 0 \<longleftrightarrow> a \<le> b" 

787 
"a  b \<le> c \<longleftrightarrow> a \<le> c + b" 

788 
"a  b \<ge> c \<longleftrightarrow> a \<ge> c + b" 

789 
by arith+ 

33175  790 

54070  791 
lemma open_ball [intro, simp]: "open (ball x e)" 
792 
proof  

793 
have "open (dist x ` {..<e})" 

794 
by (intro open_vimage open_lessThan continuous_on_intros) 

795 
also have "dist x ` {..<e} = ball x e" 

796 
by auto 

797 
finally show ?thesis . 

798 
qed 

33175  799 

800 
lemma open_contains_ball: "open S \<longleftrightarrow> (\<forall>x\<in>S. \<exists>e>0. ball x e \<subseteq> S)" 

801 
unfolding open_dist subset_eq mem_ball Ball_def dist_commute .. 

802 

33714
eb2574ac4173
Added new lemmas to Euclidean Space by Robert Himmelmann
hoelzl
parents:
33324
diff
changeset

803 
lemma openE[elim?]: 
53282  804 
assumes "open S" "x\<in>S" 
33714
eb2574ac4173
Added new lemmas to Euclidean Space by Robert Himmelmann
hoelzl
parents:
33324
diff
changeset

805 
obtains e where "e>0" "ball x e \<subseteq> S" 
eb2574ac4173
Added new lemmas to Euclidean Space by Robert Himmelmann
hoelzl
parents:
33324
diff
changeset

806 
using assms unfolding open_contains_ball by auto 
eb2574ac4173
Added new lemmas to Euclidean Space by Robert Himmelmann
hoelzl
parents:
33324
diff
changeset

807 

33175  808 
lemma open_contains_ball_eq: "open S \<Longrightarrow> \<forall>x. x\<in>S \<longleftrightarrow> (\<exists>e>0. ball x e \<subseteq> S)" 
809 
by (metis open_contains_ball subset_eq centre_in_ball) 

810 

811 
lemma ball_eq_empty[simp]: "ball x e = {} \<longleftrightarrow> e \<le> 0" 

39302
d7728f65b353
renamed lemmas: ext_iff > fun_eq_iff, set_ext_iff > set_eq_iff, set_ext > set_eqI
nipkow
parents:
39198
diff
changeset

812 
unfolding mem_ball set_eq_iff 
33175  813 
apply (simp add: not_less) 
52624  814 
apply (metis zero_le_dist order_trans dist_self) 
815 
done 

33175  816 

53291  817 
lemma ball_empty[intro]: "e \<le> 0 \<Longrightarrow> ball x e = {}" by simp 
33175  818 

50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50324
diff
changeset

819 
lemma euclidean_dist_l2: 
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50324
diff
changeset

820 
fixes x y :: "'a :: euclidean_space" 
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50324
diff
changeset

821 
shows "dist x y = setL2 (\<lambda>i. dist (x \<bullet> i) (y \<bullet> i)) Basis" 
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50324
diff
changeset

822 
unfolding dist_norm norm_eq_sqrt_inner setL2_def 
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50324
diff
changeset

823 
by (subst euclidean_inner) (simp add: power2_eq_square inner_diff_left) 
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50324
diff
changeset

824 

54775
2d3df8633dad
prefer box over greaterThanLessThan on euclidean_space
immler
parents:
54489
diff
changeset

825 
definition (in euclidean_space) eucl_less (infix "<e" 50) 
2d3df8633dad
prefer box over greaterThanLessThan on euclidean_space
immler
parents:
54489
diff
changeset

826 
where "eucl_less a b \<longleftrightarrow> (\<forall>i\<in>Basis. a \<bullet> i < b \<bullet> i)" 
2d3df8633dad
prefer box over greaterThanLessThan on euclidean_space
immler
parents:
54489
diff
changeset

827 

2d3df8633dad
prefer box over greaterThanLessThan on euclidean_space
immler
parents:
54489
diff
changeset

828 
definition box_eucl_less: "box a b = {x. a <e x \<and> x <e b}" 
56188  829 
definition "cbox a b = {x. \<forall>i\<in>Basis. a \<bullet> i \<le> x \<bullet> i \<and> x \<bullet> i \<le> b \<bullet> i}" 
54775
2d3df8633dad
prefer box over greaterThanLessThan on euclidean_space
immler
parents:
54489
diff
changeset

830 

2d3df8633dad
prefer box over greaterThanLessThan on euclidean_space
immler
parents:
54489
diff
changeset

831 
lemma box_def: "box a b = {x. \<forall>i\<in>Basis. a \<bullet> i < x \<bullet> i \<and> x \<bullet> i < b \<bullet> i}" 
2d3df8633dad
prefer box over greaterThanLessThan on euclidean_space
immler
parents:
54489
diff
changeset

832 
and in_box_eucl_less: "x \<in> box a b \<longleftrightarrow> a <e x \<and> x <e b" 
56188  833 
and mem_box: "x \<in> box a b \<longleftrightarrow> (\<forall>i\<in>Basis. a \<bullet> i < x \<bullet> i \<and> x \<bullet> i < b \<bullet> i)" 
834 
"x \<in> cbox a b \<longleftrightarrow> (\<forall>i\<in>Basis. a \<bullet> i \<le> x \<bullet> i \<and> x \<bullet> i \<le> b \<bullet> i)" 

835 
by (auto simp: box_eucl_less eucl_less_def cbox_def) 

836 

837 
lemma mem_box_real[simp]: 

838 
"(x::real) \<in> box a b \<longleftrightarrow> a < x \<and> x < b" 

839 
"(x::real) \<in> cbox a b \<longleftrightarrow> a \<le> x \<and> x \<le> b" 

840 
by (auto simp: mem_box) 

841 

842 
lemma box_real[simp]: 

843 
fixes a b:: real 

844 
shows "box a b = {a <..< b}" "cbox a b = {a .. b}" 

845 
by auto 

50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50324
diff
changeset

846 

50087  847 
lemma rational_boxes: 
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50324
diff
changeset

848 
fixes x :: "'a\<Colon>euclidean_space" 
53291  849 
assumes "e > 0" 
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50324
diff
changeset

850 
shows "\<exists>a b. (\<forall>i\<in>Basis. a \<bullet> i \<in> \<rat> \<and> b \<bullet> i \<in> \<rat> ) \<and> x \<in> box a b \<and> box a b \<subseteq> ball x e" 
50087  851 
proof  
852 
def e' \<equiv> "e / (2 * sqrt (real (DIM ('a))))" 

53291  853 
then have e: "e' > 0" 
53255  854 
using assms by (auto intro!: divide_pos_pos simp: DIM_positive) 
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50324
diff
changeset

855 
have "\<forall>i. \<exists>y. y \<in> \<rat> \<and> y < x \<bullet> i \<and> x \<bullet> i  y < e'" (is "\<forall>i. ?th i") 
50087  856 
proof 
53255  857 
fix i 
858 
from Rats_dense_in_real[of "x \<bullet> i  e'" "x \<bullet> i"] e 

859 
show "?th i" by auto 

50087  860 
qed 
55522  861 
from choice[OF this] obtain a where 
862 
a: "\<forall>xa. a xa \<in> \<rat> \<and> a xa < x \<bullet> xa \<and> x \<bullet> xa  a xa < e'" .. 

50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50324
diff
changeset

863 
have "\<forall>i. \<exists>y. y \<in> \<rat> \<and> x \<bullet> i < y \<and> y  x \<bullet> i < e'" (is "\<forall>i. ?th i") 
50087  864 
proof 
53255  865 
fix i 
866 
from Rats_dense_in_real[of "x \<bullet> i" "x \<bullet> i + e'"] e 

867 
show "?th i" by auto 

50087  868 
qed 
55522  869 
from choice[OF this] obtain b where 
870 
b: "\<forall>xa. b xa \<in> \<rat> \<and> x \<bullet> xa < b xa \<and> b xa  x \<bullet> xa < e'" .. 

50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50324
diff
changeset

871 
let ?a = "\<Sum>i\<in>Basis. a i *\<^sub>R i" and ?b = "\<Sum>i\<in>Basis. b i *\<^sub>R i" 
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50324
diff
changeset

872 
show ?thesis 
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50324
diff
changeset

873 
proof (rule exI[of _ ?a], rule exI[of _ ?b], safe) 
53255  874 
fix y :: 'a 
875 
assume *: "y \<in> box ?a ?b" 

53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
52625
diff
changeset

876 
have "dist x y = sqrt (\<Sum>i\<in>Basis. (dist (x \<bullet> i) (y \<bullet> i))\<^sup>2)" 
50087  877 
unfolding setL2_def[symmetric] by (rule euclidean_dist_l2) 
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50324
diff
changeset

878 
also have "\<dots> < sqrt (\<Sum>(i::'a)\<in>Basis. e^2 / real (DIM('a)))" 
50087  879 
proof (rule real_sqrt_less_mono, rule setsum_strict_mono) 
53255  880 
fix i :: "'a" 
881 
assume i: "i \<in> Basis" 

882 
have "a i < y\<bullet>i \<and> y\<bullet>i < b i" 

883 
using * i by (auto simp: box_def) 

884 
moreover have "a i < x\<bullet>i" "x\<bullet>i  a i < e'" 

885 
using a by auto 

886 
moreover have "x\<bullet>i < b i" "b i  x\<bullet>i < e'" 

887 
using b by auto 

888 
ultimately have "\<bar>x\<bullet>i  y\<bullet>i\<bar> < 2 * e'" 

889 
by auto 

50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50324
diff
changeset

890 
then have "dist (x \<bullet> i) (y \<bullet> i) < e/sqrt (real (DIM('a)))" 
50087  891 
unfolding e'_def by (auto simp: dist_real_def) 
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
52625
diff
changeset

892 
then have "(dist (x \<bullet> i) (y \<bullet> i))\<^sup>2 < (e/sqrt (real (DIM('a))))\<^sup>2" 
50087  893 
by (rule power_strict_mono) auto 
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
52625
diff
changeset

894 
then show "(dist (x \<bullet> i) (y \<bullet> i))\<^sup>2 < e\<^sup>2 / real DIM('a)" 
50087  895 
by (simp add: power_divide) 
896 
qed auto 

53255  897 
also have "\<dots> = e" 
898 
using `0 < e` by (simp add: real_eq_of_nat) 

899 
finally show "y \<in> ball x e" 

900 
by (auto simp: ball_def) 

50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50324
diff
changeset

901 
qed (insert a b, auto simp: box_def) 
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50324
diff
changeset

902 
qed 
51103  903 

50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50324
diff
changeset

904 
lemma open_UNION_box: 
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50324
diff
changeset

905 
fixes M :: "'a\<Colon>euclidean_space set" 
53282  906 
assumes "open M" 
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50324
diff
changeset

907 
defines "a' \<equiv> \<lambda>f :: 'a \<Rightarrow> real \<times> real. (\<Sum>(i::'a)\<in>Basis. fst (f i) *\<^sub>R i)" 
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50324
diff
changeset

908 
defines "b' \<equiv> \<lambda>f :: 'a \<Rightarrow> real \<times> real. (\<Sum>(i::'a)\<in>Basis. snd (f i) *\<^sub>R i)" 
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
52625
diff
changeset

909 
defines "I \<equiv> {f\<in>Basis \<rightarrow>\<^sub>E \<rat> \<times> \<rat>. box (a' f) (b' f) \<subseteq> M}" 
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50324
diff
changeset

910 
shows "M = (\<Union>f\<in>I. box (a' f) (b' f))" 
52624  911 
proof  
912 
{ 

913 
fix x assume "x \<in> M" 

914 
obtain e where e: "e > 0" "ball x e \<subseteq> M" 

915 
using openE[OF `open M` `x \<in> M`] by auto 

53282  916 
moreover obtain a b where ab: 
917 
"x \<in> box a b" 

918 
"\<forall>i \<in> Basis. a \<bullet> i \<in> \<rat>" 

919 
"\<forall>i\<in>Basis. b \<bullet> i \<in> \<rat>" 

920 
"box a b \<subseteq> ball x e" 

52624  921 
using rational_boxes[OF e(1)] by metis 
922 
ultimately have "x \<in> (\<Union>f\<in>I. box (a' f) (b' f))" 

923 
by (intro UN_I[of "\<lambda>i\<in>Basis. (a \<bullet> i, b \<bullet> i)"]) 

924 
(auto simp: euclidean_representation I_def a'_def b'_def) 

925 
} 

926 
then show ?thesis by (auto simp: I_def) 

927 
qed 

928 

33175  929 

930 
subsection{* Connectedness *} 

931 

932 
lemma connected_local: 

53255  933 
"connected S \<longleftrightarrow> 
934 
\<not> (\<exists>e1 e2. 

935 
openin (subtopology euclidean S) e1 \<and> 

936 
openin (subtopology euclidean S) e2 \<and> 

937 
S \<subseteq> e1 \<union> e2 \<and> 

938 
e1 \<inter> e2 = {} \<and> 

939 
e1 \<noteq> {} \<and> 

940 
e2 \<noteq> {})" 

53282  941 
unfolding connected_def openin_open 
55775  942 
by blast 
33175  943 

34105  944 
lemma exists_diff: 
945 
fixes P :: "'a set \<Rightarrow> bool" 

946 
shows "(\<exists>S. P( S)) \<longleftrightarrow> (\<exists>S. P S)" (is "?lhs \<longleftrightarrow> ?rhs") 

53255  947 
proof  
948 
{ 

949 
assume "?lhs" 

950 
then have ?rhs by blast 

951 
} 

33175  952 
moreover 
53255  953 
{ 
954 
fix S 

955 
assume H: "P S" 

34105  956 
have "S =  ( S)" by auto 
53255  957 
with H have "P ( ( S))" by metis 
958 
} 

33175  959 
ultimately show ?thesis by metis 
960 
qed 

961 

962 
lemma connected_clopen: "connected S \<longleftrightarrow> 

53255  963 
(\<forall>T. openin (subtopology euclidean S) T \<and> 
964 
closedin (subtopology euclidean S) T \<longrightarrow> T = {} \<or> T = S)" (is "?lhs \<longleftrightarrow> ?rhs") 

965 
proof  

966 
have "\<not> connected S \<longleftrightarrow> 

967 
(\<exists>e1 e2. open e1 \<and> open ( e2) \<and> S \<subseteq> e1 \<union> ( e2) \<and> e1 \<inter> ( e2) \<inter> S = {} \<and> e1 \<inter> S \<noteq> {} \<and> ( e2) \<inter> S \<noteq> {})" 

33175  968 
unfolding connected_def openin_open closedin_closed 
55775  969 
by (metis double_complement) 
53282  970 
then have th0: "connected S \<longleftrightarrow> 
53255  971 
\<not> (\<exists>e2 e1. closed e2 \<and> open e1 \<and> S \<subseteq> e1 \<union> ( e2) \<and> e1 \<inter> ( e2) \<inter> S = {} \<and> e1 \<inter> S \<noteq> {} \<and> ( e2) \<inter> S \<noteq> {})" 
52624  972 
(is " _ \<longleftrightarrow> \<not> (\<exists>e2 e1. ?P e2 e1)") 
973 
apply (simp add: closed_def) 

974 
apply metis 

975 
done 

33175  976 
have th1: "?rhs \<longleftrightarrow> \<not> (\<exists>t' t. closed t'\<and>t = S\<inter>t' \<and> t\<noteq>{} \<and> t\<noteq>S \<and> (\<exists>t'. open t' \<and> t = S \<inter> t'))" 
977 
(is "_ \<longleftrightarrow> \<not> (\<exists>t' t. ?Q t' t)") 

978 
unfolding connected_def openin_open closedin_closed by auto 

53255  979 
{ 
980 
fix e2 

981 
{ 

982 
fix e1 

53282  983 
have "?P e2 e1 \<longleftrightarrow> (\<exists>t. closed e2 \<and> t = S\<inter>e2 \<and> open e1 \<and> t = S\<inter>e1 \<and> t\<noteq>{} \<and> t \<noteq> S)" 
53255  984 
by auto 
985 
} 

986 
then have "(\<exists>e1. ?P e2 e1) \<longleftrightarrow> (\<exists>t. ?Q e2 t)" 

987 
by metis 

988 
} 

989 
then have "\<forall>e2. (\<exists>e1. ?P e2 e1) \<longleftrightarrow> (\<exists>t. ?Q e2 t)" 

990 
by blast 

991 
then show ?thesis 

992 
unfolding th0 th1 by simp 

33175  993 
qed 
994 

44210
eba74571833b
Topology_Euclidean_Space.thy: organize section headings
huffman
parents:
44207
diff
changeset

995 

33175  996 
subsection{* Limit points *} 
997 

53282  998 
definition (in topological_space) islimpt:: "'a \<Rightarrow> 'a set \<Rightarrow> bool" (infixr "islimpt" 60) 
53255  999 
where "x islimpt S \<longleftrightarrow> (\<forall>T. x\<in>T \<longrightarrow> open T \<longrightarrow> (\<exists>y\<in>S. y\<in>T \<and> y\<noteq>x))" 
33175  1000 

1001 
lemma islimptI: 

1002 
assumes "\<And>T. x \<in> T \<Longrightarrow> open T \<Longrightarrow> \<exists>y\<in>S. y \<in> T \<and> y \<noteq> x" 

1003 
shows "x islimpt S" 

1004 
using assms unfolding islimpt_def by auto 

1005 

1006 
lemma islimptE: 

1007 
assumes "x islimpt S" and "x \<in> T" and "open T" 

1008 
obtains y where "y \<in> S" and "y \<in> T" and "y \<noteq> x" 

1009 
using assms unfolding islimpt_def by auto 

1010 

44584  1011 
lemma islimpt_iff_eventually: "x islimpt S \<longleftrightarrow> \<not> eventually (\<lambda>y. y \<notin> S) (at x)" 
1012 
unfolding islimpt_def eventually_at_topological by auto 

1013 

53255  1014 
lemma islimpt_subset: "x islimpt S \<Longrightarrow> S \<subseteq> T \<Longrightarrow> x islimpt T" 
44584  1015 
unfolding islimpt_def by fast 
33175  1016 

1017 
lemma islimpt_approachable: 

1018 
fixes x :: "'a::metric_space" 

1019 
shows "x islimpt S \<longleftrightarrow> (\<forall>e>0. \<exists>x'\<in>S. x' \<noteq> x \<and> dist x' x < e)" 

44584  1020 
unfolding islimpt_iff_eventually eventually_at by fast 
33175  1021 

1022 
lemma islimpt_approachable_le: 

1023 
fixes x :: "'a::metric_space" 

53640  1024 
shows "x islimpt S \<longleftrightarrow> (\<forall>e>0. \<exists>x'\<in> S. x' \<noteq> x \<and> dist x' x \<le> e)" 
33175  1025 
unfolding islimpt_approachable 
44584  1026 
using approachable_lt_le [where f="\<lambda>y. dist y x" and P="\<lambda>y. y \<notin> S \<or> y = x", 
1027 
THEN arg_cong [where f=Not]] 

1028 
by (simp add: Bex_def conj_commute conj_left_commute) 

33175  1029 

44571  1030 
lemma islimpt_UNIV_iff: "x islimpt UNIV \<longleftrightarrow> \<not> open {x}" 
1031 
unfolding islimpt_def by (safe, fast, case_tac "T = {x}", fast, fast) 

1032 

51351  1033 
lemma islimpt_punctured: "x islimpt S = x islimpt (S{x})" 
1034 
unfolding islimpt_def by blast 

1035 

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1036 
text {* A perfect space has no isolated points. *} 
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1037 

44571  1038 
lemma islimpt_UNIV [simp, intro]: "(x::'a::perfect_space) islimpt UNIV" 
1039 
unfolding islimpt_UNIV_iff by (rule not_open_singleton) 

33175  1040 

1041 
lemma perfect_choose_dist: 

44072
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class perfect_space inherits from topological_space;
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1042 
fixes x :: "'a::{perfect_space, metric_space}" 
33175  1043 
shows "0 < r \<Longrightarrow> \<exists>a. a \<noteq> x \<and> dist a x < r" 
53255  1044 
using islimpt_UNIV [of x] 
1045 
by (simp add: islimpt_approachable) 

33175  1046 

1047 
lemma closed_limpt: "closed S \<longleftrightarrow> (\<forall>x. x islimpt S \<longrightarrow> x \<in> S)" 

1048 
unfolding closed_def 

1049 
apply (subst open_subopen) 

34105  1050 
apply (simp add: islimpt_def subset_eq) 
52624  1051 
apply (metis ComplE ComplI) 
1052 
done 

33175  1053 

1054 
lemma islimpt_EMPTY[simp]: "\<not> x islimpt {}" 

1055 
unfolding islimpt_def by auto 

1056 

1057 
lemma finite_set_avoid: 

1058 
fixes a :: "'a::metric_space" 

53255  1059 
assumes fS: "finite S" 
53640  1060 
shows "\<exists>d>0. \<forall>x\<in>S. x \<noteq> a \<longrightarrow> d \<le> dist a x" 
53255  1061 
proof (induct rule: finite_induct[OF fS]) 
1062 
case 1 

1063 
then show ?case by (auto intro: zero_less_one) 

33175  1064 
next 
1065 
case (2 x F) 

53255  1066 
from 2 obtain d where d: "d >0" "\<forall>x\<in>F. x\<noteq>a \<longrightarrow> d \<le> dist a x" 
1067 
by blast 

1068 
show ?case 

1069 
proof (cases "x = a") 

1070 
case True 

1071 
then show ?thesis using d by auto 

1072 
next 

1073 
case False 

33175  1074 
let ?d = "min d (dist a x)" 
53255  1075 
have dp: "?d > 0" 
1076 
using False d(1) using dist_nz by auto 

1077 
from d have d': "\<forall>x\<in>F. x\<noteq>a \<longrightarrow> ?d \<le> dist a x" 

1078 
by auto 

1079 
with dp False show ?thesis 

1080 
by (auto intro!: exI[where x="?d"]) 

1081 
qed 

33175  1082 
qed 
1083 

1084 
lemma islimpt_Un: "x islimpt (S \<union> T) \<longleftrightarrow> x islimpt S \<or> x islimpt T" 

50897
078590669527
generalize lemma islimpt_finite to class t1_space
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1085 
by (simp add: islimpt_iff_eventually eventually_conj_iff) 
33175  1086 

1087 
lemma discrete_imp_closed: 

1088 
fixes S :: "'a::metric_space set" 

53255  1089 
assumes e: "0 < e" 
1090 
and d: "\<forall>x \<in> S. \<forall>y \<in> S. dist y x < e \<longrightarrow> y = x" 

33175  1091 
shows "closed S" 
53255  1092 
proof  
1093 
{ 

1094 
fix x 

1095 
assume C: "\<forall>e>0. \<exists>x'\<in>S. x' \<noteq> x \<and> dist x' x < e" 

33175  1096 
from e have e2: "e/2 > 0" by arith 
53282  1097 
from C[rule_format, OF e2] obtain y where y: "y \<in> S" "y \<noteq> x" "dist y x < e/2" 
53255  1098 
by blast 
33175  1099 
let ?m = "min (e/2) (dist x y) " 
53255  1100 
from e2 y(2) have mp: "?m > 0" 
53291  1101 
by (simp add: dist_nz[symmetric]) 
53282  1102 
from C[rule_format, OF mp] obtain z where z: "z \<in> S" "z \<noteq> x" "dist z x < ?m" 
53255  1103 
by blast 
33175  1104 
have th: "dist z y < e" using z y 
1105 
by (intro dist_triangle_lt [where z=x], simp) 

1106 
from d[rule_format, OF y(1) z(1) th] y z 

1107 
have False by (auto simp add: dist_commute)} 

53255  1108 
then show ?thesis 
1109 
by (metis islimpt_approachable closed_limpt [where 'a='a]) 

33175  1110 
qed 
1111 

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1112 

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1113 
subsection {* Interior of a Set *} 
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1114 

44519  1115 
definition "interior S = \<Union>{T. open T \<and> T \<subseteq> S}" 
1116 

1117 
lemma interiorI [intro?]: 

1118 
assumes "open T" and "x \<in> T" and "T \<subseteq> S" 

1119 
shows "x \<in> interior S" 

1120 
using assms unfolding interior_def by fast 

1121 

1122 
lemma interiorE [elim?]: 

1123 
assumes "x \<in> interior S" 

1124 
obtains T where "open T" and "x \<in> T" and "T \<subseteq> S" 

1125 
using assms unfolding interior_def by fast 

1126 

1127 
lemma open_interior [simp, intro]: "open (interior S)" 

1128 
by (simp add: interior_def open_Union) 

1129 

1130 
lemma interior_subset: "interior S \<subseteq> S" 

1131 
by (auto simp add: interior_def) 

1132 

1133 
lemma interior_maximal: "T \<subseteq> S \<Longrightarrow> open T \<Longrightarrow> T \<subseteq> interior S" 

1134 
by (auto simp add: interior_def) 

1135 

1136 
lemma interior_open: "open S \<Longrightarrow> interior S = S" 

1137 
by (intro equalityI interior_subset interior_maximal subset_refl) 

33175  1138 

1139 
lemma interior_eq: "interior S = S \<longleftrightarrow> open S" 

44519  1140 
by (metis open_interior interior_open) 
1141 

1142 