Two new theories containing material ported from HOL Light about abstract topology
(* Author: L C Paulson, University of Cambridge [ported from HOL Light, metric.ml] *)
section \<open>$F$-Sigma and $G$-Delta sets in a Topological Space\<close>
text \<open>An $F$-sigma set is a countable union of closed sets; a $G$-delta set is the dual.\<close>
theory FSigma
imports Abstract_Topology
begin
definition fsigma_in
where "fsigma_in X \<equiv> countable union_of closedin X"
definition gdelta_in
where "gdelta_in X \<equiv> (countable intersection_of openin X) relative_to topspace X"
lemma fsigma_in_ascending:
"fsigma_in X S \<longleftrightarrow> (\<exists>C. (\<forall>n. closedin X (C n)) \<and> (\<forall>n. C n \<subseteq> C(Suc n)) \<and> \<Union> (range C) = S)"
unfolding fsigma_in_def
by (metis closedin_Un countable_union_of_ascending closedin_empty)
lemma gdelta_in_alt:
"gdelta_in X S \<longleftrightarrow> S \<subseteq> topspace X \<and> (countable intersection_of openin X) S"
proof -
have "(countable intersection_of openin X) (topspace X)"
by (simp add: countable_intersection_of_inc)
then show ?thesis
unfolding gdelta_in_def
by (metis countable_intersection_of_inter relative_to_def relative_to_imp_subset relative_to_subset)
qed
lemma fsigma_in_subset: "fsigma_in X S \<Longrightarrow> S \<subseteq> topspace X"
using closedin_subset by (fastforce simp: fsigma_in_def union_of_def subset_iff)
lemma gdelta_in_subset: "gdelta_in X S \<Longrightarrow> S \<subseteq> topspace X"
by (simp add: gdelta_in_alt)
lemma closed_imp_fsigma_in: "closedin X S \<Longrightarrow> fsigma_in X S"
by (simp add: countable_union_of_inc fsigma_in_def)
lemma open_imp_gdelta_in: "openin X S \<Longrightarrow> gdelta_in X S"
by (simp add: countable_intersection_of_inc gdelta_in_alt openin_subset)
lemma fsigma_in_empty [iff]: "fsigma_in X {}"
by (simp add: closed_imp_fsigma_in)
lemma gdelta_in_empty [iff]: "gdelta_in X {}"
by (simp add: open_imp_gdelta_in)
lemma fsigma_in_topspace [iff]: "fsigma_in X (topspace X)"
by (simp add: closed_imp_fsigma_in)
lemma gdelta_in_topspace [iff]: "gdelta_in X (topspace X)"
by (simp add: open_imp_gdelta_in)
lemma fsigma_in_Union:
"\<lbrakk>countable T; \<And>S. S \<in> T \<Longrightarrow> fsigma_in X S\<rbrakk> \<Longrightarrow> fsigma_in X (\<Union> T)"
by (simp add: countable_union_of_Union fsigma_in_def)
lemma fsigma_in_Un:
"\<lbrakk>fsigma_in X S; fsigma_in X T\<rbrakk> \<Longrightarrow> fsigma_in X (S \<union> T)"
by (simp add: countable_union_of_Un fsigma_in_def)
lemma fsigma_in_Int:
"\<lbrakk>fsigma_in X S; fsigma_in X T\<rbrakk> \<Longrightarrow> fsigma_in X (S \<inter> T)"
by (simp add: closedin_Int countable_union_of_Int fsigma_in_def)
lemma gdelta_in_Inter:
"\<lbrakk>countable T; T\<noteq>{}; \<And>S. S \<in> T \<Longrightarrow> gdelta_in X S\<rbrakk> \<Longrightarrow> gdelta_in X (\<Inter> T)"
by (simp add: Inf_less_eq countable_intersection_of_Inter gdelta_in_alt)
lemma gdelta_in_Int:
"\<lbrakk>gdelta_in X S; gdelta_in X T\<rbrakk> \<Longrightarrow> gdelta_in X (S \<inter> T)"
by (simp add: countable_intersection_of_inter gdelta_in_alt le_infI2)
lemma gdelta_in_Un:
"\<lbrakk>gdelta_in X S; gdelta_in X T\<rbrakk> \<Longrightarrow> gdelta_in X (S \<union> T)"
by (simp add: countable_intersection_of_union gdelta_in_alt openin_Un)
lemma fsigma_in_diff:
assumes "fsigma_in X S" "gdelta_in X T"
shows "fsigma_in X (S - T)"
proof -
have [simp]: "S - (S \<inter> T) = S - T" for S T :: "'a set"
by auto
have [simp]: "topspace X - \<Inter>\<T> = (\<Union>T\<in>\<T>. topspace X - T)" for \<T>
by auto
have "\<And>\<T>. \<lbrakk>countable \<T>; \<T> \<subseteq> Collect (openin X)\<rbrakk> \<Longrightarrow>
(countable union_of closedin X) (\<Union> ((-) (topspace X) ` \<T>))"
by (metis Ball_Collect countable_union_of_UN countable_union_of_inc openin_closedin_eq)
then have "\<forall>S. gdelta_in X S \<longrightarrow> fsigma_in X (topspace X - S)"
by (simp add: fsigma_in_def gdelta_in_def all_relative_to all_intersection_of del: UN_simps)
then show ?thesis
by (metis Diff_Int2 Diff_Int_distrib2 assms fsigma_in_Int fsigma_in_subset inf.absorb_iff2)
qed
lemma gdelta_in_diff:
assumes "gdelta_in X S" "fsigma_in X T"
shows "gdelta_in X (S - T)"
proof -
have [simp]: "topspace X - \<Union>\<T> = topspace X \<inter> (\<Inter>T\<in>\<T>. topspace X - T)" for \<T>
by auto
have "\<And>\<T>. \<lbrakk>countable \<T>; \<T> \<subseteq> Collect (closedin X)\<rbrakk>
\<Longrightarrow> (countable intersection_of openin X relative_to topspace X)
(topspace X \<inter> \<Inter> ((-) (topspace X) ` \<T>))"
by (metis Ball_Collect closedin_def countable_intersection_of_INT countable_intersection_of_inc relative_to_inc)
then have "\<forall>S. fsigma_in X S \<longrightarrow> gdelta_in X (topspace X - S)"
by (simp add: fsigma_in_def gdelta_in_def all_union_of del: INT_simps)
then show ?thesis
by (metis Diff_Int2 Diff_Int_distrib2 assms gdelta_in_Int gdelta_in_alt inf.orderE inf_commute)
qed
lemma gdelta_in_fsigma_in:
"gdelta_in X S \<longleftrightarrow> S \<subseteq> topspace X \<and> fsigma_in X (topspace X - S)"
by (metis double_diff fsigma_in_diff fsigma_in_topspace gdelta_in_alt gdelta_in_diff gdelta_in_topspace)
lemma fsigma_in_gdelta_in:
"fsigma_in X S \<longleftrightarrow> S \<subseteq> topspace X \<and> gdelta_in X (topspace X - S)"
by (metis Diff_Diff_Int fsigma_in_subset gdelta_in_fsigma_in inf.absorb_iff2)
lemma gdelta_in_descending:
"gdelta_in X S \<longleftrightarrow> (\<exists>C. (\<forall>n. openin X (C n)) \<and> (\<forall>n. C(Suc n) \<subseteq> C n) \<and> \<Inter>(range C) = S)" (is "?lhs=?rhs")
proof
assume ?lhs
then obtain C where C: "S \<subseteq> topspace X" "\<And>n. closedin X (C n)"
"\<And>n. C n \<subseteq> C(Suc n)" "\<Union>(range C) = topspace X - S"
by (meson fsigma_in_ascending gdelta_in_fsigma_in)
define D where "D \<equiv> \<lambda>n. topspace X - C n"
have "\<And>n. openin X (D n) \<and> D (Suc n) \<subseteq> D n"
by (simp add: Diff_mono C D_def openin_diff)
moreover have "\<Inter>(range D) = S"
by (simp add: Diff_Diff_Int Int_absorb1 C D_def)
ultimately show ?rhs
by metis
next
assume ?rhs
then obtain C where "S \<subseteq> topspace X"
and C: "\<And>n. openin X (C n)" "\<And>n. C(Suc n) \<subseteq> C n" "\<Inter>(range C) = S"
using openin_subset by fastforce
define D where "D \<equiv> \<lambda>n. topspace X - C n"
have "\<And>n. closedin X (D n) \<and> D n \<subseteq> D(Suc n)"
by (simp add: Diff_mono C closedin_diff D_def)
moreover have "\<Union>(range D) = topspace X - S"
using C D_def by blast
ultimately show ?lhs
by (metis \<open>S \<subseteq> topspace X\<close> fsigma_in_ascending gdelta_in_fsigma_in)
qed
lemma homeomorphic_map_fsigmaness:
assumes f: "homeomorphic_map X Y f" and "U \<subseteq> topspace X"
shows "fsigma_in Y (f ` U) \<longleftrightarrow> fsigma_in X U" (is "?lhs=?rhs")
proof -
obtain g where g: "homeomorphic_maps X Y f g" and g: "homeomorphic_map Y X g"
and 1: "(\<forall>x \<in> topspace X. g(f x) = x)" and 2: "(\<forall>y \<in> topspace Y. f(g y) = y)"
using assms homeomorphic_map_maps by (metis homeomorphic_maps_map)
show ?thesis
proof
assume ?lhs
then obtain \<V> where "countable \<V>" and \<V>: "\<V> \<subseteq> Collect (closedin Y)" "\<Union>\<V> = f`U"
by (force simp: fsigma_in_def union_of_def)
define \<U> where "\<U> \<equiv> image (image g) \<V>"
have "countable \<U>"
using \<U>_def \<open>countable \<V>\<close> by blast
moreover have "\<U> \<subseteq> Collect (closedin X)"
using f g homeomorphic_map_closedness_eq \<V> unfolding \<U>_def by blast
moreover have "\<Union>\<U> \<subseteq> U"
unfolding \<U>_def by (smt (verit) assms 1 \<V> image_Union image_iff in_mono subsetI)
moreover have "U \<subseteq> \<Union>\<U>"
unfolding \<U>_def using assms \<V>
by (smt (verit, del_insts) "1" imageI image_Union subset_iff)
ultimately show ?rhs
by (metis fsigma_in_def subset_antisym union_of_def)
next
assume ?rhs
then obtain \<V> where "countable \<V>" and \<V>: "\<V> \<subseteq> Collect (closedin X)" "\<Union>\<V> = U"
by (auto simp: fsigma_in_def union_of_def)
define \<U> where "\<U> \<equiv> image (image f) \<V>"
have "countable \<U>"
using \<U>_def \<open>countable \<V>\<close> by blast
moreover have "\<U> \<subseteq> Collect (closedin Y)"
using f g homeomorphic_map_closedness_eq \<V> unfolding \<U>_def by blast
moreover have "\<Union>\<U> = f`U"
unfolding \<U>_def using \<V> by blast
ultimately show ?lhs
by (metis fsigma_in_def union_of_def)
qed
qed
lemma homeomorphic_map_fsigmaness_eq:
"homeomorphic_map X Y f
\<Longrightarrow> (fsigma_in X U \<longleftrightarrow> U \<subseteq> topspace X \<and> fsigma_in Y (f ` U))"
by (metis fsigma_in_subset homeomorphic_map_fsigmaness)
lemma homeomorphic_map_gdeltaness:
assumes "homeomorphic_map X Y f" "U \<subseteq> topspace X"
shows "gdelta_in Y (f ` U) \<longleftrightarrow> gdelta_in X U"
proof -
have "topspace Y - f ` U = f ` (topspace X - U)"
by (metis (no_types, lifting) Diff_subset assms homeomorphic_eq_everything_map inj_on_image_set_diff)
then show ?thesis
using assms homeomorphic_imp_surjective_map
by (fastforce simp: gdelta_in_fsigma_in homeomorphic_map_fsigmaness_eq)
qed
lemma homeomorphic_map_gdeltaness_eq:
"homeomorphic_map X Y f
\<Longrightarrow> gdelta_in X U \<longleftrightarrow> U \<subseteq> topspace X \<and> gdelta_in Y (f ` U)"
by (meson gdelta_in_subset homeomorphic_map_gdeltaness)
lemma fsigma_in_relative_to:
"(fsigma_in X relative_to S) = fsigma_in (subtopology X S)"
by (simp add: fsigma_in_def countable_union_of_relative_to closedin_relative_to)
lemma fsigma_in_subtopology:
"fsigma_in (subtopology X U) S \<longleftrightarrow> (\<exists>T. fsigma_in X T \<and> S = T \<inter> U)"
by (metis fsigma_in_relative_to inf_commute relative_to_def)
lemma gdelta_in_relative_to:
"(gdelta_in X relative_to S) = gdelta_in (subtopology X S)"
apply (simp add: gdelta_in_def)
by (metis countable_intersection_of_relative_to openin_relative_to subtopology_restrict)
lemma gdelta_in_subtopology:
"gdelta_in (subtopology X U) S \<longleftrightarrow> (\<exists>T. gdelta_in X T \<and> S = T \<inter> U)"
by (metis gdelta_in_relative_to inf_commute relative_to_def)
lemma fsigma_in_fsigma_subtopology:
"fsigma_in X S \<Longrightarrow> (fsigma_in (subtopology X S) T \<longleftrightarrow> fsigma_in X T \<and> T \<subseteq> S)"
by (metis fsigma_in_Int fsigma_in_gdelta_in fsigma_in_subtopology inf.orderE topspace_subtopology_subset)
lemma gdelta_in_gdelta_subtopology:
"gdelta_in X S \<Longrightarrow> (gdelta_in (subtopology X S) T \<longleftrightarrow> gdelta_in X T \<and> T \<subseteq> S)"
by (metis gdelta_in_Int gdelta_in_subset gdelta_in_subtopology inf.orderE topspace_subtopology_subset)
end