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Datei: Lindelof_Spaces.thy Sprache: Isabelle

Original von: Isabelle^©

section\<open>Lindel\"of spaces\<close>

theory Lindelof_Spaces
imports T1_Spaces
begin

definition Lindelof_space where
 "Lindelof_space X \
 \<forall>\. (\<forall>U \<in> \. openin X U) \<and> \<Union>\ = topspace X
 \<longrightarrow> (\<exists>\<V>. countable \<V> \<and> \<V> \<subseteq> \ \<and> \<Union>\<V> = topspace X)"

lemma Lindelof_spaceD:
 "\Lindelof_space X; \U. U \ \ \ openin X U; \\ = topspace X\
 \<Longrightarrow> \<exists>\<V>. countable \<V> \<and> \<V> \<subseteq> \ \<and> \<Union>\<V> = topspace X"
 by (auto simp: Lindelof_space_def)

lemma Lindelof_space_alt:
 "Lindelof_space X \
 (\<forall>\. (\<forall>U \<in> \. openin X U) \<and> topspace X \<subseteq> \<Union>\
 \<longrightarrow> (\<exists>\<V>. countable \<V> \<and> \<V> \<subseteq> \ \<and> topspace X \<subseteq> \<Union>\<V>))"
 unfolding Lindelof_space_def
 using openin_subset by fastforce

lemma compact_imp_Lindelof_space:
 "compact_space X \ Lindelof_space X"
 unfolding Lindelof_space_def compact_space
 by (meson uncountable_infinite)

lemma Lindelof_space_topspace_empty:
 "topspace X = {} \ Lindelof_space X"
 using compact_imp_Lindelof_space compact_space_topspace_empty by blast

lemma Lindelof_space_Union:
 assumes \: "countable \" and lin: "\<And>U. U \<in> \ \<Longrightarrow> Lindelof_space (subtopology X U)"
 shows "Lindelof_space (subtopology X (\\))"
proof -
 have "\\. countable \ \ \ \ \ \ \\ \ \\ = topspace X \ \\"
 if \<F>: "\<F> \<subseteq> Collect (openin X)" and UF: "\<Union>\ \<inter> \<Union>\<F> = topspace X \<inter> \<Union>\"
 for \<F>
 proof -
 have "\U. \U \ \; U \ \\ = topspace X \ U\
 \<Longrightarrow> \<exists>\<V>. countable \<V> \<and> \<V> \<subseteq> \<F> \<and> U \<inter> \<Union>\<V> = topspace X \<inter> U"
 using lin \<F>
 unfolding Lindelof_space_def openin_subtopology_alt Ball_def subset_iff [symmetric]
 by (simp add: all_subset_image imp_conjL ex_countable_subset_image)
 then obtain g where g: "\U. \U \ \; U \ \\ = topspace X \ U\
 \<Longrightarrow> countable (g U) \<and> (g U) \<subseteq> \<F> \<and> U \<inter> \<Union>(g U) = topspace X \<inter> U"
 by metis
 show ?thesis
 proof (intro exI conjI)
 show "countable (\(g ` \))"
 using Int_commute UF g by (fastforce intro: countable_UN [OF \])
 show "\(g ` \) \ \"
 using g UF by blast
 show "\\ \ \(\(g ` \)) = topspace X \ \\"
 proof
 show "\\ \ \(\(g ` \)) \ topspace X \ \\"
 using g UF by blast
 show "topspace X \ \\ \ \\ \ \(\(g ` \))"
 proof clarsimp
 show "\y\\. \W\g y. x \ W"
 if "x \ topspace X" "x \ V" "V \ \" for x V
 proof -
 have "V \ \\ = topspace X \ V"
 using UF \<open>V \<in> \\<close> by blast
 with that g [OF \<open>V \<in> \\<close>] show ?thesis by blast
 qed
 qed
 qed
 qed
 qed
 then show ?thesis
 unfolding Lindelof_space_def openin_subtopology_alt Ball_def subset_iff [symmetric]
 by (simp add: all_subset_image imp_conjL ex_countable_subset_image)
qed

lemma countable_imp_Lindelof_space:
 assumes "countable(topspace X)"
 shows "Lindelof_space X"
proof -
 have "Lindelof_space (subtopology X (\x \ topspace X. {x}))"
 proof (rule Lindelof_space_Union)
 show "countable ((\x. {x}) ` topspace X)"
 using assms by blast
 show "Lindelof_space (subtopology X U)"
 if "U \ (\x. {x}) ` topspace X" for U
 proof -
 have "compactin X U"
 using that by force
 then show ?thesis
 by (meson compact_imp_Lindelof_space compact_space_subtopology)
 qed
 qed
 then show ?thesis
 by simp
qed
lemma Lindelof_space_subtopology:
 "Lindelof_space(subtopology X S) \
 (\<forall>\. (\<forall>U \<in> \. openin X U) \<and> topspace X \<inter> S \<subseteq> \<Union>\
 \<longrightarrow> (\<exists>V. countable V \<and> V \<subseteq> \ \<and> topspace X \<inter> S \<subseteq> \<Union>V))"
proof -
 have *: "(S \ \\ = topspace X \ S) = (topspace X \ S \ \\)"
 if "\x. x \ \ \ openin X x" for \
 by (blast dest: openin_subset [OF that])
 moreover have "(\ \ \ \ S \ \\ = topspace X \ S) = (\ \ \ \ topspace X \ S \ \\)"
 if "\x. x \ \ \ openin X x" "topspace X \ S \ \\" "countable \" for \ \
 using that * by blast
 ultimately show ?thesis
 unfolding Lindelof_space_def openin_subtopology_alt Ball_def
 apply (simp add: all_subset_image imp_conjL ex_countable_subset_image flip: subset_iff)
 apply (intro all_cong1 imp_cong ex_cong, auto)
 done
qed

lemma Lindelof_space_subtopology_subset:
 "S \ topspace X
 \<Longrightarrow> (Lindelof_space(subtopology X S) \<longleftrightarrow>
 (\<forall>\. (\<forall>U \<in> \. openin X U) \<and> S \<subseteq> \<Union>\
 \<longrightarrow> (\<exists>V. countable V \<and> V \<subseteq> \ \<and> S \<subseteq> \<Union>V)))"
 by (metis Lindelof_space_subtopology topspace_subtopology topspace_subtopology_subset)

lemma Lindelof_space_closedin_subtopology:
 assumes X: "Lindelof_space X" and clo: "closedin X S"
 shows "Lindelof_space (subtopology X S)"
proof -
 have "S \ topspace X"
 by (simp add: clo closedin_subset)
 then show ?thesis
 proof (clarsimp simp add: Lindelof_space_subtopology_subset)
 show "\V. countable V \ V \ \ \ S \ \V"
 if "\U\\. openin X U" and "S \ \\" for \
 proof -
 have "\\. countable \ \ \ \ insert (topspace X - S) \ \ \\ = topspace X"
 proof (rule Lindelof_spaceD [OF X, of "insert (topspace X - S) \"])
 show "openin X U"
 if "U \ insert (topspace X - S) \" for U
 using that \<open>\<forall>U\<in>\<F>. openin X U\<close> clo by blast
 show "\(insert (topspace X - S) \) = topspace X"
 apply auto
 apply (meson in_mono openin_closedin_eq that(1))
 using UnionE \<open>S \<subseteq> \<Union>\<F>\<close> by auto
 qed
 then obtain \<V> where "countable \<V>" "\<V> \<subseteq> insert (topspace X - S) \<F>" "\<Union>\<V> = topspace X"
 by metis
 with \<open>S \<subseteq> topspace X\<close>
 show ?thesis
 by (rule_tac x="(\ - {topspace X - S})" in exI) auto
 qed
 qed
qed

lemma Lindelof_space_continuous_map_image:
 assumes X: "Lindelof_space X" and f: "continuous_map X Y f" and fim: "f ` (topspace X) = topspace Y"
 shows "Lindelof_space Y"
proof -
 have "\\. countable \ \ \ \ \ \ \\ = topspace Y"
 if \: "\<And>U. U \<in> \ \<Longrightarrow> openin Y U" and UU: "\<Union>\ = topspace Y" for \
 proof -
 define \<V> where "\<V> \<equiv> (\<lambda>U. {x \<in> topspace X. f x \<in> U}) ` \"
 have "\V. V \ \ \ openin X V"
 unfolding \<V>_def using \ continuous_map f by fastforce
 moreover have "\\ = topspace X"
 unfolding \<V>_def using UU fim by fastforce
 ultimately have "\\. countable \ \ \ \ \ \ \\ = topspace X"
 using X by (simp add: Lindelof_space_def)
 then obtain \<C> where "countable \<C>" "\<C> \<subseteq> \" and \<C>: "(\<Union>U\<in>\<C>. {x \<in> topspace X. f x \<in> U}) = topspace X"
 by (metis (no_types, lifting) \<V>_def countable_subset_image)
 moreover have "\\ = topspace Y"
 proof
 show "\\ \ topspace Y"
 using UU \<C> \<open>\<C> \<subseteq> \\<close> by fastforce
 have "y \ \\" if "y \ topspace Y" for y
 proof -
 obtain x where "x \ topspace X" "y = f x"
 using that fim by (metis \<open>y \<in> topspace Y\<close> imageE)
 with \<C> show ?thesis by auto
 qed
 then show "topspace Y \ \\" by blast
 qed
 ultimately show ?thesis
 by blast
 qed
 then show ?thesis
 unfolding Lindelof_space_def
 by auto
qed

lemma Lindelof_space_quotient_map_image:
 "\quotient_map X Y q; Lindelof_space X\ \ Lindelof_space Y"
 by (meson Lindelof_space_continuous_map_image quotient_imp_continuous_map quotient_imp_surjective_map)

lemma Lindelof_space_retraction_map_image:
 "\retraction_map X Y r; Lindelof_space X\ \ Lindelof_space Y"
 using Abstract_Topology.retraction_imp_quotient_map Lindelof_space_quotient_map_image by blast

lemma locally_finite_cover_of_Lindelof_space:
 assumes X: "Lindelof_space X" and UU: "topspace X \ \\" and fin: "locally_finite_in X \"
 shows "countable \"
proof -
 have UU_eq: "\\ = topspace X"
 by (meson UU fin locally_finite_in_def subset_antisym)
 obtain T where T: "\x. x \ topspace X \ openin X (T x) \ x \ T x \ finite {U \ \. U \ T x \ {}}"
 using fin unfolding locally_finite_in_def by metis
 then obtain I where "countable I" "I \ topspace X" and I: "topspace X \ \(T ` I)"
 using X unfolding Lindelof_space_alt
 by (drule_tac x="image T (topspace X)" in spec) (auto simp: ex_countable_subset_image)
 show ?thesis
 proof (rule countable_subset)
 have "\i. i \ I \ countable {U \ \. U \ T i \ {}}"
 using T
 by (meson \<open>I \<subseteq> topspace X\<close> in_mono uncountable_infinite)
 then show "countable (insert {} (\i\I. {U \ \. U \ T i \ {}}))"
 by (simp add: \<open>countable I\<close>)
 qed (use UU_eq I in auto)
qed

lemma Lindelof_space_proper_map_preimage:
 assumes f: "proper_map X Y f" and Y: "Lindelof_space Y"
 shows "Lindelof_space X"
proof (clarsimp simp: Lindelof_space_alt)
 show "\\. countable \ \ \ \ \ \ topspace X \ \\"
 if \: "\<forall>U\<in>\. openin X U" and sub_UU: "topspace X \<subseteq> \<Union>\" for \
 proof -
 have "\\. finite \ \ \ \ \ \ {x \ topspace X. f x = y} \ \\" if "y \ topspace Y" for y
 proof (rule compactinD)
 show "compactin X {x \ topspace X. f x = y}"
 using f proper_map_def that by fastforce
 qed (use sub_UU \ in auto)
 then obtain \<V> where \<V>: "\<And>y. y \<in> topspace Y \<Longrightarrow> finite (\<V> y) \<and> \<V> y \<subseteq> \ \<and> {x \<in> topspace X. f x = y} \<subseteq> \<Union>(\<V> y)"
 by meson
 define \<W> where "\<W> \<equiv> (\<lambda>y. topspace Y - image f (topspace X - \<Union>(\<V> y))) ` topspace Y"
 have "\U \ \. openin Y U"
 using f \ \<V> unfolding \<W>_def proper_map_def closed_map_def
 by (simp add: closedin_diff openin_Union openin_diff subset_iff)
 moreover have "topspace Y \ \\"
 using \<V> unfolding \<W>_def by clarsimp fastforce
 ultimately have "\\. countable \ \ \ \ \ \ topspace Y \ \\"
 using Y by (simp add: Lindelof_space_alt)
 then obtain I where "countable I" "I \ topspace Y"
 and I: "topspace Y \ (\i\I. topspace Y - f ` (topspace X - \(\ i)))"
 unfolding \<W>_def ex_countable_subset_image by metis
 show ?thesis
 proof (intro exI conjI)
 have "\i. i \ I \ countable (\ i)"
 by (meson \<V> \<open>I \<subseteq> topspace Y\<close> in_mono uncountable_infinite)
 with \<open>countable I\<close> show "countable (\<Union>(\<V> ` I))"
 by auto
 show "\(\ ` I) \ \"
 using \<V> \<open>I \<subseteq> topspace Y\<close> by fastforce
 show "topspace X \ \(\(\ ` I))"
 proof
 show "x \ \ (\ (\ ` I))" if "x \ topspace X" for x
 proof -
 have "f x \ topspace Y"
 by (meson f image_subset_iff proper_map_imp_subset_topspace that)
 then show ?thesis
 using that I by auto
 qed
 qed
 qed
 qed
qed

lemma Lindelof_space_perfect_map_image:
 "\Lindelof_space X; perfect_map X Y f\ \ Lindelof_space Y"
 using Lindelof_space_quotient_map_image perfect_imp_quotient_map by blast

lemma Lindelof_space_perfect_map_image_eq:
 "perfect_map X Y f \ Lindelof_space X \ Lindelof_space Y"
 using Lindelof_space_perfect_map_image Lindelof_space_proper_map_preimage perfect_map_def by blast

end

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