(* Author: L C Paulson, University of Cambridge [ported from HOL Light] *)
section \<open>Various Forms of Topological Spaces\<close>
theory Abstract_Topological_Spaces
imports Lindelof_Spaces Locally Abstract_Euclidean_Space Sum_Topology FSigma
begin
subsection\<open>Connected topological spaces\<close>
lemma connected_space_eq_frontier_eq_empty:
"connected_space X \<longleftrightarrow> (\<forall>S. S \<subseteq> topspace X \<and> X frontier_of S = {} \<longrightarrow> S = {} \<or> S = topspace X)"
by (meson clopenin_eq_frontier_of connected_space_clopen_in)
lemma connected_space_frontier_eq_empty:
"connected_space X \<and> S \<subseteq> topspace X
\<Longrightarrow> (X frontier_of S = {} \<longleftrightarrow> S = {} \<or> S = topspace X)"
by (meson connected_space_eq_frontier_eq_empty frontier_of_empty frontier_of_topspace)
lemma connectedin_eq_subset_separated_union:
"connectedin X C \<longleftrightarrow>
C \<subseteq> topspace X \<and> (\<forall>S T. separatedin X S T \<and> C \<subseteq> S \<union> T \<longrightarrow> C \<subseteq> S \<or> C \<subseteq> T)" (is "?lhs=?rhs")
proof
assume ?lhs then show ?rhs
using connectedin_subset_topspace connectedin_subset_separated_union by blast
next
assume ?rhs
then show ?lhs
by (metis closure_of_subset connectedin_separation dual_order.eq_iff inf.orderE separatedin_def sup.boundedE)
qed
lemma connectedin_clopen_cases:
"\<lbrakk>connectedin X C; closedin X T; openin X T\<rbrakk> \<Longrightarrow> C \<subseteq> T \<or> disjnt C T"
by (metis Diff_eq_empty_iff Int_empty_right clopenin_eq_frontier_of connectedin_Int_frontier_of disjnt_def)
lemma connected_space_retraction_map_image:
"\<lbrakk>retraction_map X X' r; connected_space X\<rbrakk> \<Longrightarrow> connected_space X'"
using connected_space_quotient_map_image retraction_imp_quotient_map by blast
lemma connectedin_imp_perfect_gen:
assumes X: "t1_space X" and S: "connectedin X S" and nontriv: "\<nexists>a. S = {a}"
shows "S \<subseteq> X derived_set_of S"
unfolding derived_set_of_def
proof (intro subsetI CollectI conjI strip)
show XS: "x \<in> topspace X" if "x \<in> S" for x
using that S connectedin by fastforce
show "\<exists>y. y \<noteq> x \<and> y \<in> S \<and> y \<in> T"
if "x \<in> S" and "x \<in> T \<and> openin X T" for x T
proof -
have opeXx: "openin X (topspace X - {x})"
by (meson X openin_topspace t1_space_openin_delete_alt)
moreover
have "S \<subseteq> T \<union> (topspace X - {x})"
using XS that(2) by auto
moreover have "(topspace X - {x}) \<inter> S \<noteq> {}"
by (metis Diff_triv S connectedin double_diff empty_subsetI inf_commute insert_subsetI nontriv that(1))
ultimately show ?thesis
using that connectedinD [OF S, of T "topspace X - {x}"]
by blast
qed
qed
lemma connectedin_imp_perfect:
"\<lbrakk>Hausdorff_space X; connectedin X S; \<nexists>a. S = {a}\<rbrakk> \<Longrightarrow> S \<subseteq> X derived_set_of S"
by (simp add: Hausdorff_imp_t1_space connectedin_imp_perfect_gen)
subsection\<open>The notion of "separated between" (complement of "connected between)"\<close>
definition separated_between
where "separated_between X S T \<equiv>
\<exists>U V. openin X U \<and> openin X V \<and> U \<union> V = topspace X \<and> disjnt U V \<and> S \<subseteq> U \<and> T \<subseteq> V"
lemma separated_between_alt:
"separated_between X S T \<longleftrightarrow>
(\<exists>U V. closedin X U \<and> closedin X V \<and> U \<union> V = topspace X \<and> disjnt U V \<and> S \<subseteq> U \<and> T \<subseteq> V)"
unfolding separated_between_def
by (metis separatedin_open_sets separation_closedin_Un_gen subtopology_topspace
separatedin_closed_sets separation_openin_Un_gen)
lemma separated_between:
"separated_between X S T \<longleftrightarrow>
(\<exists>U. closedin X U \<and> openin X U \<and> S \<subseteq> U \<and> T \<subseteq> topspace X - U)"
unfolding separated_between_def closedin_def disjnt_def
by (smt (verit, del_insts) Diff_cancel Diff_disjoint Diff_partition Un_Diff Un_Diff_Int openin_subset)
lemma separated_between_mono:
"\<lbrakk>separated_between X S T; S' \<subseteq> S; T' \<subseteq> T\<rbrakk> \<Longrightarrow> separated_between X S' T'"
by (meson order.trans separated_between)
lemma separated_between_refl:
"separated_between X S S \<longleftrightarrow> S = {}"
unfolding separated_between_def
by (metis Un_empty_right disjnt_def disjnt_empty2 disjnt_subset2 disjnt_sym le_iff_inf openin_empty openin_topspace)
lemma separated_between_sym:
"separated_between X S T \<longleftrightarrow> separated_between X T S"
by (metis disjnt_sym separated_between_alt sup_commute)
lemma separated_between_imp_subset:
"separated_between X S T \<Longrightarrow> S \<subseteq> topspace X \<and> T \<subseteq> topspace X"
by (metis le_supI1 le_supI2 separated_between_def)
lemma separated_between_empty:
"(separated_between X {} S \<longleftrightarrow> S \<subseteq> topspace X) \<and> (separated_between X S {} \<longleftrightarrow> S \<subseteq> topspace X)"
by (metis Diff_empty bot.extremum closedin_empty openin_empty separated_between separated_between_imp_subset separated_between_sym)
lemma separated_between_Un:
"separated_between X S (T \<union> U) \<longleftrightarrow> separated_between X S T \<and> separated_between X S U"
by (auto simp: separated_between)
lemma separated_between_Un':
"separated_between X (S \<union> T) U \<longleftrightarrow> separated_between X S U \<and> separated_between X T U"
by (simp add: separated_between_Un separated_between_sym)
lemma separated_between_imp_disjoint:
"separated_between X S T \<Longrightarrow> disjnt S T"
by (meson disjnt_iff separated_between_def subsetD)
lemma separated_between_imp_separatedin:
"separated_between X S T \<Longrightarrow> separatedin X S T"
by (meson separated_between_def separatedin_mono separatedin_open_sets)
lemma separated_between_full:
assumes "S \<union> T = topspace X"
shows "separated_between X S T \<longleftrightarrow> disjnt S T \<and> closedin X S \<and> openin X S \<and> closedin X T \<and> openin X T"
proof -
have "separated_between X S T \<longrightarrow> separatedin X S T"
by (simp add: separated_between_imp_separatedin)
then show ?thesis
unfolding separated_between_def
by (metis assms separation_closedin_Un_gen separation_openin_Un_gen subset_refl subtopology_topspace)
qed
lemma separated_between_eq_separatedin:
"S \<union> T = topspace X \<Longrightarrow> (separated_between X S T \<longleftrightarrow> separatedin X S T)"
by (simp add: separated_between_full separatedin_full)
lemma separated_between_pointwise_left:
assumes "compactin X S"
shows "separated_between X S T \<longleftrightarrow>
(S = {} \<longrightarrow> T \<subseteq> topspace X) \<and> (\<forall>x \<in> S. separated_between X {x} T)" (is "?lhs=?rhs")
proof
assume ?lhs then show ?rhs
using separated_between_imp_subset separated_between_mono by fastforce
next
assume R: ?rhs
then have "T \<subseteq> topspace X"
by (meson equals0I separated_between_imp_subset)
show ?lhs
proof -
obtain U where U: "\<forall>x \<in> S. openin X (U x)"
"\<forall>x \<in> S. \<exists>V. openin X V \<and> U x \<union> V = topspace X \<and> disjnt (U x) V \<and> {x} \<subseteq> U x \<and> T \<subseteq> V"
using R unfolding separated_between_def by metis
then have "S \<subseteq> \<Union>(U ` S)"
by blast
then obtain K where "finite K" "K \<subseteq> S" and K: "S \<subseteq> (\<Union>i\<in>K. U i)"
using assms U unfolding compactin_def by (smt (verit) finite_subset_image imageE)
show ?thesis
unfolding separated_between
proof (intro conjI exI)
have "\<And>x. x \<in> K \<Longrightarrow> closedin X (U x)"
by (smt (verit) \<open>K \<subseteq> S\<close> Diff_cancel U(2) Un_Diff Un_Diff_Int disjnt_def openin_closedin_eq subsetD)
then show "closedin X (\<Union> (U ` K))"
by (metis (mono_tags, lifting) \<open>finite K\<close> closedin_Union finite_imageI image_iff)
show "openin X (\<Union> (U ` K))"
using U(1) \<open>K \<subseteq> S\<close> by blast
show "S \<subseteq> \<Union> (U ` K)"
by (simp add: K)
have "\<And>x i. \<lbrakk>x \<in> T; i \<in> K; x \<in> U i\<rbrakk> \<Longrightarrow> False"
by (meson U(2) \<open>K \<subseteq> S\<close> disjnt_iff subsetD)
then show "T \<subseteq> topspace X - \<Union> (U ` K)"
using \<open>T \<subseteq> topspace X\<close> by auto
qed
qed
qed
lemma separated_between_pointwise_right:
"compactin X T
\<Longrightarrow> separated_between X S T \<longleftrightarrow> (T = {} \<longrightarrow> S \<subseteq> topspace X) \<and> (\<forall>y \<in> T. separated_between X S {y})"
by (meson separated_between_pointwise_left separated_between_sym)
lemma separated_between_closure_of:
"S \<subseteq> topspace X \<Longrightarrow> separated_between X (X closure_of S) T \<longleftrightarrow> separated_between X S T"
by (meson closure_of_minimal_eq separated_between_alt)
lemma separated_between_closure_of':
"T \<subseteq> topspace X \<Longrightarrow> separated_between X S (X closure_of T) \<longleftrightarrow> separated_between X S T"
by (meson separated_between_closure_of separated_between_sym)
lemma separated_between_closure_of_eq:
"separated_between X S T \<longleftrightarrow> S \<subseteq> topspace X \<and> separated_between X (X closure_of S) T"
by (metis separated_between_closure_of separated_between_imp_subset)
lemma separated_between_closure_of_eq':
"separated_between X S T \<longleftrightarrow> T \<subseteq> topspace X \<and> separated_between X S (X closure_of T)"
by (metis separated_between_closure_of' separated_between_imp_subset)
lemma separated_between_frontier_of_eq':
"separated_between X S T \<longleftrightarrow>
T \<subseteq> topspace X \<and> disjnt S T \<and> separated_between X S (X frontier_of T)" (is "?lhs=?rhs")
proof
assume ?lhs then show ?rhs
by (metis interior_of_union_frontier_of separated_between_Un separated_between_closure_of_eq'
separated_between_imp_disjoint)
next
assume R: ?rhs
then obtain U where U: "closedin X U" "openin X U" "S \<subseteq> U" "X closure_of T - X interior_of T \<subseteq> topspace X - U"
by (metis frontier_of_def separated_between)
show ?lhs
proof (rule separated_between_mono [of _ S "X closure_of T"])
have "separated_between X S T"
unfolding separated_between
proof (intro conjI exI)
show "S \<subseteq> U - T" "T \<subseteq> topspace X - (U - T)"
using R U(3) by (force simp: disjnt_iff)+
have "T \<subseteq> X closure_of T"
by (simp add: R closure_of_subset)
then have *: "U - T = U - X interior_of T"
using U(4) interior_of_subset by fastforce
then show "closedin X (U - T)"
by (simp add: U(1) closedin_diff)
have "U \<inter> X frontier_of T = {}"
using U(4) frontier_of_def by fastforce
then show "openin X (U - T)"
by (metis * Diff_Un U(2) Un_Diff_Int Un_Int_eq(1) closedin_closure_of interior_of_union_frontier_of openin_diff sup_bot_right)
qed
then show "separated_between X S (X closure_of T)"
by (simp add: R separated_between_closure_of')
qed (auto simp: R closure_of_subset)
qed
lemma separated_between_frontier_of_eq:
"separated_between X S T \<longleftrightarrow> S \<subseteq> topspace X \<and> disjnt S T \<and> separated_between X (X frontier_of S) T"
by (metis disjnt_sym separated_between_frontier_of_eq' separated_between_sym)
lemma separated_between_frontier_of:
"\<lbrakk>S \<subseteq> topspace X; disjnt S T\<rbrakk>
\<Longrightarrow> (separated_between X (X frontier_of S) T \<longleftrightarrow> separated_between X S T)"
using separated_between_frontier_of_eq by blast
lemma separated_between_frontier_of':
"\<lbrakk>T \<subseteq> topspace X; disjnt S T\<rbrakk>
\<Longrightarrow> (separated_between X S (X frontier_of T) \<longleftrightarrow> separated_between X S T)"
using separated_between_frontier_of_eq' by auto
lemma connected_space_separated_between:
"connected_space X \<longleftrightarrow> (\<forall>S T. separated_between X S T \<longrightarrow> S = {} \<or> T = {})" (is "?lhs=?rhs")
proof
assume ?lhs then show ?rhs
by (metis Diff_cancel connected_space_clopen_in separated_between subset_empty)
next
assume ?rhs then show ?lhs
by (meson connected_space_eq_not_separated separated_between_eq_separatedin)
qed
lemma connected_space_imp_separated_between_trivial:
"connected_space X
\<Longrightarrow> (separated_between X S T \<longleftrightarrow> S = {} \<and> T \<subseteq> topspace X \<or> S \<subseteq> topspace X \<and> T = {})"
by (metis connected_space_separated_between separated_between_empty)
subsection\<open>Connected components\<close>
lemma connected_component_of_subtopology_eq:
"connected_component_of (subtopology X U) a = connected_component_of X a \<longleftrightarrow>
connected_component_of_set X a \<subseteq> U"
by (force simp: connected_component_of_set connectedin_subtopology connected_component_of_def fun_eq_iff subset_iff)
lemma connected_components_of_subtopology:
assumes "C \<in> connected_components_of X" "C \<subseteq> U"
shows "C \<in> connected_components_of (subtopology X U)"
proof -
obtain a where a: "connected_component_of_set X a \<subseteq> U" and "a \<in> topspace X"
and Ceq: "C = connected_component_of_set X a"
using assms by (force simp: connected_components_of_def)
then have "a \<in> U"
by (simp add: connected_component_of_refl in_mono)
then have "connected_component_of_set X a = connected_component_of_set (subtopology X U) a"
by (metis a connected_component_of_subtopology_eq)
then show ?thesis
by (simp add: Ceq \<open>a \<in> U\<close> \<open>a \<in> topspace X\<close> connected_component_in_connected_components_of)
qed
lemma open_in_finite_connected_components:
assumes "finite(connected_components_of X)" "C \<in> connected_components_of X"
shows "openin X C"
proof -
have "closedin X (topspace X - C)"
by (metis DiffD1 assms closedin_Union closedin_connected_components_of complement_connected_components_of_Union finite_Diff)
then show ?thesis
by (simp add: assms connected_components_of_subset openin_closedin)
qed
thm connected_component_of_eq_overlap
lemma connected_components_of_disjoint:
assumes "C \<in> connected_components_of X" "C' \<in> connected_components_of X"
shows "(disjnt C C' \<longleftrightarrow> (C \<noteq> C'))"
using assms nonempty_connected_components_of pairwiseD pairwise_disjoint_connected_components_of by fastforce
lemma connected_components_of_overlap:
"\<lbrakk>C \<in> connected_components_of X; C' \<in> connected_components_of X\<rbrakk> \<Longrightarrow> C \<inter> C' \<noteq> {} \<longleftrightarrow> C = C'"
by (meson connected_components_of_disjoint disjnt_def)
lemma pairwise_separated_connected_components_of:
"pairwise (separatedin X) (connected_components_of X)"
by (simp add: closedin_connected_components_of connected_components_of_disjoint pairwiseI separatedin_closed_sets)
lemma finite_connected_components_of_finite:
"finite(topspace X) \<Longrightarrow> finite(connected_components_of X)"
by (simp add: Union_connected_components_of finite_UnionD)
lemma connected_component_of_unique:
"\<lbrakk>x \<in> C; connectedin X C; \<And>C'. x \<in> C' \<and> connectedin X C' \<Longrightarrow> C' \<subseteq> C\<rbrakk>
\<Longrightarrow> connected_component_of_set X x = C"
by (meson connected_component_of_maximal connectedin_connected_component_of subsetD subset_antisym)
lemma closedin_connected_component_of_subtopology:
"\<lbrakk>C \<in> connected_components_of (subtopology X s); X closure_of C \<subseteq> s\<rbrakk> \<Longrightarrow> closedin X C"
by (metis closedin_Int_closure_of closedin_connected_components_of closure_of_eq inf.absorb_iff2)
lemma connected_component_of_discrete_topology:
"connected_component_of_set (discrete_topology U) x = (if x \<in> U then {x} else {})"
by (simp add: locally_path_connected_space_discrete_topology flip: path_component_eq_connected_component_of)
lemma connected_components_of_discrete_topology:
"connected_components_of (discrete_topology U) = (\<lambda>x. {x}) ` U"
by (simp add: connected_component_of_discrete_topology connected_components_of_def)
lemma connected_component_of_continuous_image:
"\<lbrakk>continuous_map X Y f; connected_component_of X x y\<rbrakk>
\<Longrightarrow> connected_component_of Y (f x) (f y)"
by (meson connected_component_of_def connectedin_continuous_map_image image_eqI)
lemma homeomorphic_map_connected_component_of:
assumes "homeomorphic_map X Y f" and x: "x \<in> topspace X"
shows "connected_component_of_set Y (f x) = f ` (connected_component_of_set X x)"
proof -
obtain g where g: "continuous_map X Y f"
"continuous_map Y X g " "\<And>x. x \<in> topspace X \<Longrightarrow> g (f x) = x"
"\<And>y. y \<in> topspace Y \<Longrightarrow> f (g y) = y"
using assms(1) homeomorphic_map_maps homeomorphic_maps_def by fastforce
show ?thesis
using connected_component_in_topspace [of Y] x g
connected_component_of_continuous_image [of X Y f]
connected_component_of_continuous_image [of Y X g]
by force
qed
lemma homeomorphic_map_connected_components_of:
assumes "homeomorphic_map X Y f"
shows "connected_components_of Y = (image f) ` (connected_components_of X)"
proof -
have "topspace Y = f ` topspace X"
by (metis assms homeomorphic_imp_surjective_map)
with homeomorphic_map_connected_component_of [OF assms] show ?thesis
by (auto simp: connected_components_of_def image_iff)
qed
lemma connected_component_of_pair:
"connected_component_of_set (prod_topology X Y) (x,y) =
connected_component_of_set X x \<times> connected_component_of_set Y y"
proof (cases "x \<in> topspace X \<and> y \<in> topspace Y")
case True
show ?thesis
proof (rule connected_component_of_unique)
show "(x, y) \<in> connected_component_of_set X x \<times> connected_component_of_set Y y"
using True by (simp add: connected_component_of_refl)
show "connectedin (prod_topology X Y) (connected_component_of_set X x \<times> connected_component_of_set Y y)"
by (metis connectedin_Times connectedin_connected_component_of)
show "C \<subseteq> connected_component_of_set X x \<times> connected_component_of_set Y y"
if "(x, y) \<in> C \<and> connectedin (prod_topology X Y) C" for C
using that unfolding connected_component_of_def
apply clarsimp
by (metis (no_types) connectedin_continuous_map_image continuous_map_fst continuous_map_snd fst_conv imageI snd_conv)
qed
next
case False then show ?thesis
by (metis Sigma_empty1 Sigma_empty2 connected_component_of_eq_empty mem_Sigma_iff topspace_prod_topology)
qed
lemma connected_components_of_prod_topology:
"connected_components_of (prod_topology X Y) =
{C \<times> D |C D. C \<in> connected_components_of X \<and> D \<in> connected_components_of Y}" (is "?lhs=?rhs")
proof
show "?lhs \<subseteq> ?rhs"
apply (clarsimp simp: connected_components_of_def)
by (metis (no_types) connected_component_of_pair imageI)
next
show "?rhs \<subseteq> ?lhs"
using connected_component_of_pair
by (fastforce simp: connected_components_of_def)
qed
lemma connected_component_of_product_topology:
"connected_component_of_set (product_topology X I) x =
(if x \<in> extensional I then PiE I (\<lambda>i. connected_component_of_set (X i) (x i)) else {})"
(is "?lhs = If _ ?R _")
proof (cases "x \<in> topspace(product_topology X I)")
case True
have "?lhs = (\<Pi>\<^sub>E i\<in>I. connected_component_of_set (X i) (x i))"
if xX: "\<And>i. i\<in>I \<Longrightarrow> x i \<in> topspace (X i)" and ext: "x \<in> extensional I"
proof (rule connected_component_of_unique)
show "x \<in> ?R"
by (simp add: PiE_iff connected_component_of_refl local.ext xX)
show "connectedin (product_topology X I) ?R"
by (simp add: connectedin_PiE connectedin_connected_component_of)
show "C \<subseteq> ?R"
if "x \<in> C \<and> connectedin (product_topology X I) C" for C
proof -
have "C \<subseteq> extensional I"
using PiE_def connectedin_subset_topspace that by fastforce
have "\<And>y. y \<in> C \<Longrightarrow> y \<in> (\<Pi> i\<in>I. connected_component_of_set (X i) (x i))"
apply (simp add: connected_component_of_def Pi_def)
by (metis connectedin_continuous_map_image continuous_map_product_projection imageI that)
then show ?thesis
using PiE_def \<open>C \<subseteq> extensional I\<close> by fastforce
qed
qed
with True show ?thesis
by (simp add: PiE_iff)
next
case False
then show ?thesis
by (smt (verit, best) PiE_eq_empty_iff PiE_iff connected_component_of_eq_empty topspace_product_topology)
qed
lemma connected_components_of_product_topology:
"connected_components_of (product_topology X I) =
{PiE I B |B. \<forall>i \<in> I. B i \<in> connected_components_of(X i)}" (is "?lhs=?rhs")
proof
show "?lhs \<subseteq> ?rhs"
by (auto simp: connected_components_of_def connected_component_of_product_topology PiE_iff)
show "?rhs \<subseteq> ?lhs"
proof
fix F
assume "F \<in> ?rhs"
then obtain B where Feq: "F = Pi\<^sub>E I B" and
"\<forall>i\<in>I. \<exists>x\<in>topspace (X i). B i = connected_component_of_set (X i) x"
by (force simp: connected_components_of_def connected_component_of_product_topology image_iff)
then obtain f where
f: "\<And>i. i \<in> I \<Longrightarrow> f i \<in> topspace (X i) \<and> B i = connected_component_of_set (X i) (f i)"
by metis
then have "(\<lambda>i\<in>I. f i) \<in> ((\<Pi>\<^sub>E i\<in>I. topspace (X i)) \<inter> extensional I)"
by simp
with f show "F \<in> ?lhs"
unfolding Feq connected_components_of_def connected_component_of_product_topology image_iff
by (smt (verit, del_insts) PiE_cong restrict_PiE_iff restrict_apply' restrict_extensional topspace_product_topology)
qed
qed
subsection \<open>Monotone maps (in the general topological sense)\<close>
definition monotone_map
where "monotone_map X Y f ==
f ` (topspace X) \<subseteq> topspace Y \<and>
(\<forall>y \<in> topspace Y. connectedin X {x \<in> topspace X. f x = y})"
lemma monotone_map:
"monotone_map X Y f \<longleftrightarrow>
f ` (topspace X) \<subseteq> topspace Y \<and> (\<forall>y. connectedin X {x \<in> topspace X. f x = y})"
apply (simp add: monotone_map_def)
by (metis (mono_tags, lifting) connectedin_empty [of X] Collect_empty_eq image_subset_iff)
lemma monotone_map_in_subtopology:
"monotone_map X (subtopology Y S) f \<longleftrightarrow> monotone_map X Y f \<and> f ` (topspace X) \<subseteq> S"
by (smt (verit, del_insts) le_inf_iff monotone_map topspace_subtopology)
lemma monotone_map_from_subtopology:
assumes "monotone_map X Y f"
"\<And>x y. \<lbrakk>x \<in> topspace X; y \<in> topspace X; x \<in> S; f x = f y\<rbrakk> \<Longrightarrow> y \<in> S"
shows "monotone_map (subtopology X S) Y f"
proof -
have "\<And>y. y \<in> topspace Y \<Longrightarrow> connectedin X {x \<in> topspace X. x \<in> S \<and> f x = y}"
by (smt (verit) Collect_cong assms connectedin_empty empty_def monotone_map_def)
then show ?thesis
using assms by (auto simp: monotone_map_def connectedin_subtopology)
qed
lemma monotone_map_restriction:
"monotone_map X Y f \<and> {x \<in> topspace X. f x \<in> v} = u
\<Longrightarrow> monotone_map (subtopology X u) (subtopology Y v) f"
by (smt (verit, best) IntI Int_Collect image_subset_iff mem_Collect_eq monotone_map monotone_map_from_subtopology topspace_subtopology)
lemma injective_imp_monotone_map:
assumes "f ` topspace X \<subseteq> topspace Y" "inj_on f (topspace X)"
shows "monotone_map X Y f"
unfolding monotone_map_def
proof (intro conjI assms strip)
fix y
assume "y \<in> topspace Y"
then have "{x \<in> topspace X. f x = y} = {} \<or> (\<exists>a \<in> topspace X. {x \<in> topspace X. f x = y} = {a})"
using assms(2) unfolding inj_on_def by blast
then show "connectedin X {x \<in> topspace X. f x = y}"
by (metis (no_types, lifting) connectedin_empty connectedin_sing)
qed
lemma embedding_imp_monotone_map:
"embedding_map X Y f \<Longrightarrow> monotone_map X Y f"
by (metis (no_types) embedding_map_def homeomorphic_eq_everything_map inf.absorb_iff2 injective_imp_monotone_map topspace_subtopology)
lemma section_imp_monotone_map:
"section_map X Y f \<Longrightarrow> monotone_map X Y f"
by (simp add: embedding_imp_monotone_map section_imp_embedding_map)
lemma homeomorphic_imp_monotone_map:
"homeomorphic_map X Y f \<Longrightarrow> monotone_map X Y f"
by (meson section_and_retraction_eq_homeomorphic_map section_imp_monotone_map)
proposition connected_space_monotone_quotient_map_preimage:
assumes f: "monotone_map X Y f" "quotient_map X Y f" and "connected_space Y"
shows "connected_space X"
proof (rule ccontr)
assume "\<not> connected_space X"
then obtain U V where "openin X U" "openin X V" "U \<inter> V = {}"
"U \<noteq> {}" "V \<noteq> {}" and topUV: "topspace X \<subseteq> U \<union> V"
by (auto simp: connected_space_def)
then have UVsub: "U \<subseteq> topspace X" "V \<subseteq> topspace X"
by (auto simp: openin_subset)
have "\<not> connected_space Y"
unfolding connected_space_def not_not
proof (intro exI conjI)
show "topspace Y \<subseteq> f`U \<union> f`V"
by (metis f(2) image_Un quotient_imp_surjective_map subset_Un_eq topUV)
show "f`U \<noteq> {}"
by (simp add: \<open>U \<noteq> {}\<close>)
show "(f`V) \<noteq> {}"
by (simp add: \<open>V \<noteq> {}\<close>)
have *: "y \<notin> f ` V" if "y \<in> f ` U" for y
proof -
have \<section>: "connectedin X {x \<in> topspace X. f x = y}"
using f(1) monotone_map by fastforce
show ?thesis
using connectedinD [OF \<section> \<open>openin X U\<close> \<open>openin X V\<close>] UVsub topUV \<open>U \<inter> V = {}\<close> that
by (force simp: disjoint_iff)
qed
then show "f`U \<inter> f`V = {}"
by blast
show "openin Y (f`U)"
using f \<open>openin X U\<close> topUV * unfolding quotient_map_saturated_open by force
show "openin Y (f`V)"
using f \<open>openin X V\<close> topUV * unfolding quotient_map_saturated_open by force
qed
then show False
by (simp add: assms)
qed
lemma connectedin_monotone_quotient_map_preimage:
assumes "monotone_map X Y f" "quotient_map X Y f" "connectedin Y C" "openin Y C \<or> closedin Y C"
shows "connectedin X {x \<in> topspace X. f x \<in> C}"
proof -
have "connected_space (subtopology X {x \<in> topspace X. f x \<in> C})"
proof -
have "connected_space (subtopology Y C)"
using \<open>connectedin Y C\<close> connectedin_def by blast
moreover have "quotient_map (subtopology X {a \<in> topspace X. f a \<in> C}) (subtopology Y C) f"
by (simp add: assms quotient_map_restriction)
ultimately show ?thesis
using \<open>monotone_map X Y f\<close> connected_space_monotone_quotient_map_preimage monotone_map_restriction by blast
qed
then show ?thesis
by (simp add: connectedin_def)
qed
lemma monotone_open_map:
assumes "continuous_map X Y f" "open_map X Y f" and fim: "f ` (topspace X) = topspace Y"
shows "monotone_map X Y f \<longleftrightarrow> (\<forall>C. connectedin Y C \<longrightarrow> connectedin X {x \<in> topspace X. f x \<in> C})"
(is "?lhs=?rhs")
proof
assume L: ?lhs
show ?rhs
unfolding connectedin_def
proof (intro strip conjI)
fix C
assume C: "C \<subseteq> topspace Y \<and> connected_space (subtopology Y C)"
show "connected_space (subtopology X {x \<in> topspace X. f x \<in> C})"
proof (rule connected_space_monotone_quotient_map_preimage)
show "monotone_map (subtopology X {x \<in> topspace X. f x \<in> C}) (subtopology Y C) f"
by (simp add: L monotone_map_restriction)
show "quotient_map (subtopology X {x \<in> topspace X. f x \<in> C}) (subtopology Y C) f"
proof (rule continuous_open_imp_quotient_map)
show "continuous_map (subtopology X {x \<in> topspace X. f x \<in> C}) (subtopology Y C) f"
using assms continuous_map_from_subtopology continuous_map_in_subtopology by fastforce
qed (use open_map_restriction assms in fastforce)+
qed (simp add: C)
qed auto
next
assume ?rhs
then have "\<forall>y. connectedin Y {y} \<longrightarrow> connectedin X {x \<in> topspace X. f x = y}"
by (smt (verit) Collect_cong singletonD singletonI)
then show ?lhs
by (simp add: fim monotone_map_def)
qed
lemma monotone_closed_map:
assumes "continuous_map X Y f" "closed_map X Y f" and fim: "f ` (topspace X) = topspace Y"
shows "monotone_map X Y f \<longleftrightarrow> (\<forall>C. connectedin Y C \<longrightarrow> connectedin X {x \<in> topspace X. f x \<in> C})"
(is "?lhs=?rhs")
proof
assume L: ?lhs
show ?rhs
unfolding connectedin_def
proof (intro strip conjI)
fix C
assume C: "C \<subseteq> topspace Y \<and> connected_space (subtopology Y C)"
show "connected_space (subtopology X {x \<in> topspace X. f x \<in> C})"
proof (rule connected_space_monotone_quotient_map_preimage)
show "monotone_map (subtopology X {x \<in> topspace X. f x \<in> C}) (subtopology Y C) f"
by (simp add: L monotone_map_restriction)
show "quotient_map (subtopology X {x \<in> topspace X. f x \<in> C}) (subtopology Y C) f"
proof (rule continuous_closed_imp_quotient_map)
show "continuous_map (subtopology X {x \<in> topspace X. f x \<in> C}) (subtopology Y C) f"
using assms continuous_map_from_subtopology continuous_map_in_subtopology by fastforce
qed (use closed_map_restriction assms in fastforce)+
qed (simp add: C)
qed auto
next
assume ?rhs
then have "\<forall>y. connectedin Y {y} \<longrightarrow> connectedin X {x \<in> topspace X. f x = y}"
by (smt (verit) Collect_cong singletonD singletonI)
then show ?lhs
by (simp add: fim monotone_map_def)
qed
subsection\<open>Other countability properties\<close>
definition second_countable
where "second_countable X \<equiv>
\<exists>\<B>. countable \<B> \<and> (\<forall>V \<in> \<B>. openin X V) \<and>
(\<forall>U x. openin X U \<and> x \<in> U \<longrightarrow> (\<exists>V \<in> \<B>. x \<in> V \<and> V \<subseteq> U))"
definition first_countable
where "first_countable X \<equiv>
\<forall>x \<in> topspace X.
\<exists>\<B>. countable \<B> \<and> (\<forall>V \<in> \<B>. openin X V) \<and>
(\<forall>U. openin X U \<and> x \<in> U \<longrightarrow> (\<exists>V \<in> \<B>. x \<in> V \<and> V \<subseteq> U))"
definition separable_space
where "separable_space X \<equiv>
\<exists>C. countable C \<and> C \<subseteq> topspace X \<and> X closure_of C = topspace X"
lemma second_countable:
"second_countable X \<longleftrightarrow>
(\<exists>\<B>. countable \<B> \<and> openin X = arbitrary union_of (\<lambda>x. x \<in> \<B>))"
by (smt (verit) openin_topology_base_unique second_countable_def)
lemma second_countable_subtopology:
assumes "second_countable X"
shows "second_countable (subtopology X S)"
proof -
obtain \<B> where \<B>: "countable \<B>" "\<And>V. V \<in> \<B> \<Longrightarrow> openin X V"
"\<And>U x. openin X U \<and> x \<in> U \<longrightarrow> (\<exists>V \<in> \<B>. x \<in> V \<and> V \<subseteq> U)"
using assms by (auto simp: second_countable_def)
show ?thesis
unfolding second_countable_def
proof (intro exI conjI)
show "\<forall>V\<in>((\<inter>)S) ` \<B>. openin (subtopology X S) V"
using openin_subtopology_Int2 \<B> by blast
show "\<forall>U x. openin (subtopology X S) U \<and> x \<in> U \<longrightarrow> (\<exists>V\<in>((\<inter>)S) ` \<B>. x \<in> V \<and> V \<subseteq> U)"
using \<B> unfolding openin_subtopology
by (smt (verit, del_insts) IntI image_iff inf_commute inf_le1 subset_iff)
qed (use \<B> in auto)
qed
lemma second_countable_discrete_topology:
"second_countable(discrete_topology U) \<longleftrightarrow> countable U" (is "?lhs=?rhs")
proof
assume L: ?lhs
then
obtain \<B> where \<B>: "countable \<B>" "\<And>V. V \<in> \<B> \<Longrightarrow> V \<subseteq> U"
"\<And>W x. W \<subseteq> U \<and> x \<in> W \<longrightarrow> (\<exists>V \<in> \<B>. x \<in> V \<and> V \<subseteq> W)"
by (auto simp: second_countable_def)
then have "{x} \<in> \<B>" if "x \<in> U" for x
by (metis empty_subsetI insertCI insert_subset subset_antisym that)
then show ?rhs
by (smt (verit) countable_subset image_subsetI \<open>countable \<B>\<close> countable_image_inj_on [OF _ inj_singleton])
next
assume ?rhs
then show ?lhs
unfolding second_countable_def
by (rule_tac x="(\<lambda>x. {x}) ` U" in exI) auto
qed
lemma second_countable_open_map_image:
assumes "continuous_map X Y f" "open_map X Y f"
and fim: "f ` (topspace X) = topspace Y" and "second_countable X"
shows "second_countable Y"
proof -
have openXYf: "\<And>U. openin X U \<longrightarrow> openin Y (f ` U)"
using assms by (auto simp: open_map_def)
obtain \<B> where \<B>: "countable \<B>" "\<And>V. V \<in> \<B> \<Longrightarrow> openin X V"
and *: "\<And>U x. openin X U \<and> x \<in> U \<longrightarrow> (\<exists>V \<in> \<B>. x \<in> V \<and> V \<subseteq> U)"
using assms by (auto simp: second_countable_def)
show ?thesis
unfolding second_countable_def
proof (intro exI conjI strip)
fix V y
assume V: "openin Y V \<and> y \<in> V"
then obtain x where "x \<in> topspace X" and x: "f x = y"
by (metis fim image_iff openin_subset subsetD)
then obtain W where "W\<in>\<B>" "x \<in> W" "W \<subseteq> {x \<in> topspace X. f x \<in> V}"
using * [of "{x \<in> topspace X. f x \<in> V}" x] V assms openin_continuous_map_preimage
by force
then show "\<exists>W \<in> (image f) ` \<B>. y \<in> W \<and> W \<subseteq> V"
using x by auto
qed (use \<B> openXYf in auto)
qed
lemma homeomorphic_space_second_countability:
"X homeomorphic_space Y \<Longrightarrow> (second_countable X \<longleftrightarrow> second_countable Y)"
by (meson homeomorphic_eq_everything_map homeomorphic_space homeomorphic_space_sym second_countable_open_map_image)
lemma second_countable_retraction_map_image:
"\<lbrakk>retraction_map X Y r; second_countable X\<rbrakk> \<Longrightarrow> second_countable Y"
using hereditary_imp_retractive_property homeomorphic_space_second_countability second_countable_subtopology by blast
lemma second_countable_imp_first_countable:
"second_countable X \<Longrightarrow> first_countable X"
by (metis first_countable_def second_countable_def)
lemma first_countable_subtopology:
assumes "first_countable X"
shows "first_countable (subtopology X S)"
unfolding first_countable_def
proof
fix x
assume "x \<in> topspace (subtopology X S)"
then obtain \<B> where "countable \<B>" and \<B>: "\<And>V. V \<in> \<B> \<Longrightarrow> openin X V"
"\<And>U. openin X U \<and> x \<in> U \<longrightarrow> (\<exists>V \<in> \<B>. x \<in> V \<and> V \<subseteq> U)"
using assms first_countable_def by force
show "\<exists>\<B>. countable \<B> \<and> (\<forall>V\<in>\<B>. openin (subtopology X S) V) \<and> (\<forall>U. openin (subtopology X S) U \<and> x \<in> U \<longrightarrow> (\<exists>V\<in>\<B>. x \<in> V \<and> V \<subseteq> U))"
proof (intro exI conjI strip)
show "countable (((\<inter>)S) ` \<B>)"
using \<open>countable \<B>\<close> by blast
show "openin (subtopology X S) V" if "V \<in> ((\<inter>)S) ` \<B>" for V
using \<B> openin_subtopology_Int2 that by fastforce
show "\<exists>V\<in>((\<inter>)S) ` \<B>. x \<in> V \<and> V \<subseteq> U"
if "openin (subtopology X S) U \<and> x \<in> U" for U
using that \<B>(2) by (clarsimp simp: openin_subtopology) (meson le_infI2)
qed
qed
lemma first_countable_discrete_topology:
"first_countable (discrete_topology U)"
unfolding first_countable_def topspace_discrete_topology openin_discrete_topology
proof
fix x assume "x \<in> U"
show "\<exists>\<B>. countable \<B> \<and> (\<forall>V\<in>\<B>. V \<subseteq> U) \<and> (\<forall>Ua. Ua \<subseteq> U \<and> x \<in> Ua \<longrightarrow> (\<exists>V\<in>\<B>. x \<in> V \<and> V \<subseteq> Ua))"
using \<open>x \<in> U\<close> by (rule_tac x="{{x}}" in exI) auto
qed
lemma first_countable_open_map_image:
assumes "continuous_map X Y f" "open_map X Y f"
and fim: "f ` (topspace X) = topspace Y" and "first_countable X"
shows "first_countable Y"
unfolding first_countable_def
proof
fix y
assume "y \<in> topspace Y"
have openXYf: "\<And>U. openin X U \<longrightarrow> openin Y (f ` U)"
using assms by (auto simp: open_map_def)
then obtain x where x: "x \<in> topspace X" "f x = y"
by (metis \<open>y \<in> topspace Y\<close> fim imageE)
obtain \<B> where \<B>: "countable \<B>" "\<And>V. V \<in> \<B> \<Longrightarrow> openin X V"
and *: "\<And>U. openin X U \<and> x \<in> U \<longrightarrow> (\<exists>V \<in> \<B>. x \<in> V \<and> V \<subseteq> U)"
using assms x first_countable_def by force
show "\<exists>\<B>. countable \<B> \<and>
(\<forall>V\<in>\<B>. openin Y V) \<and> (\<forall>U. openin Y U \<and> y \<in> U \<longrightarrow> (\<exists>V\<in>\<B>. y \<in> V \<and> V \<subseteq> U))"
proof (intro exI conjI strip)
fix V assume "openin Y V \<and> y \<in> V"
then have "\<exists>W\<in>\<B>. x \<in> W \<and> W \<subseteq> {x \<in> topspace X. f x \<in> V}"
using * [of "{x \<in> topspace X. f x \<in> V}"] assms openin_continuous_map_preimage x
by fastforce
then show "\<exists>V' \<in> (image f) ` \<B>. y \<in> V' \<and> V' \<subseteq> V"
using image_mono x by auto
qed (use \<B> openXYf in force)+
qed
lemma homeomorphic_space_first_countability:
"X homeomorphic_space Y \<Longrightarrow> first_countable X \<longleftrightarrow> first_countable Y"
by (meson first_countable_open_map_image homeomorphic_eq_everything_map homeomorphic_space homeomorphic_space_sym)
lemma first_countable_retraction_map_image:
"\<lbrakk>retraction_map X Y r; first_countable X\<rbrakk> \<Longrightarrow> first_countable Y"
using first_countable_subtopology hereditary_imp_retractive_property homeomorphic_space_first_countability by blast
lemma separable_space_open_subset:
assumes "separable_space X" "openin X S"
shows "separable_space (subtopology X S)"
proof -
obtain C where C: "countable C" "C \<subseteq> topspace X" "X closure_of C = topspace X"
by (meson assms separable_space_def)
then have "\<And>x T. \<lbrakk>x \<in> topspace X; x \<in> T; openin (subtopology X S) T\<rbrakk>
\<Longrightarrow> \<exists>y. y \<in> S \<and> y \<in> C \<and> y \<in> T"
by (smt (verit) \<open>openin X S\<close> in_closure_of openin_open_subtopology subsetD)
with C \<open>openin X S\<close> show ?thesis
unfolding separable_space_def
by (rule_tac x="S \<inter> C" in exI) (force simp: in_closure_of)
qed
lemma separable_space_continuous_map_image:
assumes "separable_space X" "continuous_map X Y f"
and fim: "f ` (topspace X) = topspace Y"
shows "separable_space Y"
proof -
have cont: "\<And>S. f ` (X closure_of S) \<subseteq> Y closure_of f ` S"
by (simp add: assms continuous_map_image_closure_subset)
obtain C where C: "countable C" "C \<subseteq> topspace X" "X closure_of C = topspace X"
by (meson assms separable_space_def)
then show ?thesis
unfolding separable_space_def
by (metis cont fim closure_of_subset_topspace countable_image image_mono subset_antisym)
qed
lemma separable_space_quotient_map_image:
"\<lbrakk>quotient_map X Y q; separable_space X\<rbrakk> \<Longrightarrow> separable_space Y"
by (meson quotient_imp_continuous_map quotient_imp_surjective_map separable_space_continuous_map_image)
lemma separable_space_retraction_map_image:
"\<lbrakk>retraction_map X Y r; separable_space X\<rbrakk> \<Longrightarrow> separable_space Y"
using retraction_imp_quotient_map separable_space_quotient_map_image by blast
lemma homeomorphic_separable_space:
"X homeomorphic_space Y \<Longrightarrow> (separable_space X \<longleftrightarrow> separable_space Y)"
by (meson homeomorphic_eq_everything_map homeomorphic_maps_map homeomorphic_space_def separable_space_continuous_map_image)
lemma separable_space_discrete_topology:
"separable_space(discrete_topology U) \<longleftrightarrow> countable U"
by (metis countable_Int2 discrete_topology_closure_of dual_order.refl inf.orderE separable_space_def topspace_discrete_topology)
lemma second_countable_imp_separable_space:
assumes "second_countable X"
shows "separable_space X"
proof -
obtain \<B> where \<B>: "countable \<B>" "\<And>V. V \<in> \<B> \<Longrightarrow> openin X V"
and *: "\<And>U x. openin X U \<and> x \<in> U \<longrightarrow> (\<exists>V \<in> \<B>. x \<in> V \<and> V \<subseteq> U)"
using assms by (auto simp: second_countable_def)
obtain c where c: "\<And>V. \<lbrakk>V \<in> \<B>; V \<noteq> {}\<rbrakk> \<Longrightarrow> c V \<in> V"
by (metis all_not_in_conv)
then have **: "\<And>x. x \<in> topspace X \<Longrightarrow> x \<in> X closure_of c ` (\<B> - {{}})"
using * by (force simp: closure_of_def)
show ?thesis
unfolding separable_space_def
proof (intro exI conjI)
show "countable (c ` (\<B>-{{}}))"
using \<B>(1) by blast
show "(c ` (\<B>-{{}})) \<subseteq> topspace X"
using \<B>(2) c openin_subset by fastforce
show "X closure_of (c ` (\<B>-{{}})) = topspace X"
by (meson ** closure_of_subset_topspace subsetI subset_antisym)
qed
qed
lemma second_countable_imp_Lindelof_space:
assumes "second_countable X"
shows "Lindelof_space X"
unfolding Lindelof_space_def
proof clarify
fix \<U>
assume "\<forall>U \<in> \<U>. openin X U" and UU: "\<Union>\<U> = topspace X"
obtain \<B> where \<B>: "countable \<B>" "\<And>V. V \<in> \<B> \<Longrightarrow> openin X V"
and *: "\<And>U x. openin X U \<and> x \<in> U \<longrightarrow> (\<exists>V \<in> \<B>. x \<in> V \<and> V \<subseteq> U)"
using assms by (auto simp: second_countable_def)
define \<B>' where "\<B>' = {B \<in> \<B>. \<exists>U. U \<in> \<U> \<and> B \<subseteq> U}"
have \<B>': "countable \<B>'" "\<Union>\<B>' = \<Union>\<U>"
using \<B> using "*" \<open>\<forall>U\<in>\<U>. openin X U\<close> by (fastforce simp: \<B>'_def)+
have "\<And>b. \<exists>U. b \<in> \<B>' \<longrightarrow> U \<in> \<U> \<and> b \<subseteq> U"
by (simp add: \<B>'_def)
then obtain G where G: "\<And>b. b \<in> \<B>' \<longrightarrow> G b \<in> \<U> \<and> b \<subseteq> G b"
by metis
with \<B>' UU show "\<exists>\<V>. countable \<V> \<and> \<V> \<subseteq> \<U> \<and> \<Union>\<V> = topspace X"
by (rule_tac x="G ` \<B>'" in exI) fastforce
qed
subsection \<open>Neigbourhood bases EXTRAS\<close>
text \<open>Neigbourhood bases: useful for "local" properties of various kinds\<close>
lemma openin_topology_neighbourhood_base_unique:
"openin X = arbitrary union_of P \<longleftrightarrow>
(\<forall>u. P u \<longrightarrow> openin X u) \<and> neighbourhood_base_of P X"
by (smt (verit, best) open_neighbourhood_base_of openin_topology_base_unique)
lemma neighbourhood_base_at_topology_base:
" openin X = arbitrary union_of b
\<Longrightarrow> (neighbourhood_base_at x P X \<longleftrightarrow>
(\<forall>w. b w \<and> x \<in> w \<longrightarrow> (\<exists>u v. openin X u \<and> P v \<and> x \<in> u \<and> u \<subseteq> v \<and> v \<subseteq> w)))"
apply (simp add: neighbourhood_base_at_def)
by (smt (verit, del_insts) openin_topology_base_unique subset_trans)
lemma neighbourhood_base_of_unlocalized:
assumes "\<And>S t. P S \<and> openin X t \<and> (t \<noteq> {}) \<and> t \<subseteq> S \<Longrightarrow> P t"
shows "neighbourhood_base_of P X \<longleftrightarrow>
(\<forall>x \<in> topspace X. \<exists>u v. openin X u \<and> P v \<and> x \<in> u \<and> u \<subseteq> v \<and> v \<subseteq> topspace X)"
apply (simp add: neighbourhood_base_of_def)
by (smt (verit, ccfv_SIG) assms empty_iff neighbourhood_base_at_unlocalized)
lemma neighbourhood_base_at_discrete_topology:
"neighbourhood_base_at x P (discrete_topology u) \<longleftrightarrow> x \<in> u \<Longrightarrow> P {x}"
apply (simp add: neighbourhood_base_at_def)
by (smt (verit) empty_iff empty_subsetI insert_subset singletonI subsetD subset_singletonD)
lemma neighbourhood_base_of_discrete_topology:
"neighbourhood_base_of P (discrete_topology u) \<longleftrightarrow> (\<forall>x \<in> u. P {x})"
apply (simp add: neighbourhood_base_of_def)
using neighbourhood_base_at_discrete_topology[of _ P u]
by (metis empty_subsetI insert_subset neighbourhood_base_at_def openin_discrete_topology singletonI)
lemma second_countable_neighbourhood_base_alt:
"second_countable X \<longleftrightarrow>
(\<exists>\<B>. countable \<B> \<and> (\<forall>V \<in> \<B>. openin X V) \<and> neighbourhood_base_of (\<lambda>A. A\<in>\<B>) X)"
by (metis (full_types) openin_topology_neighbourhood_base_unique second_countable)
lemma first_countable_neighbourhood_base_alt:
"first_countable X \<longleftrightarrow>
(\<forall>x \<in> topspace X. \<exists>\<B>. countable \<B> \<and> (\<forall>V \<in> \<B>. openin X V) \<and> neighbourhood_base_at x (\<lambda>V. V \<in> \<B>) X)"
unfolding first_countable_def
apply (intro ball_cong refl ex_cong conj_cong)
by (metis (mono_tags, lifting) open_neighbourhood_base_at)
lemma second_countable_neighbourhood_base:
"second_countable X \<longleftrightarrow>
(\<exists>\<B>. countable \<B> \<and> neighbourhood_base_of (\<lambda>V. V \<in> \<B>) X)" (is "?lhs=?rhs")
proof
assume ?lhs
then show ?rhs
using second_countable_neighbourhood_base_alt by blast
next
assume ?rhs
then obtain \<B> where "countable \<B>"
and \<B>: "\<And>W x. openin X W \<and> x \<in> W \<longrightarrow> (\<exists>U. openin X U \<and> (\<exists>V. V \<in> \<B> \<and> x \<in> U \<and> U \<subseteq> V \<and> V \<subseteq> W))"
by (metis neighbourhood_base_of)
then show ?lhs
unfolding second_countable_neighbourhood_base_alt neighbourhood_base_of
apply (rule_tac x="(\<lambda>u. X interior_of u) ` \<B>" in exI)
by (smt (verit, best) interior_of_eq interior_of_mono countable_image image_iff openin_interior_of)
qed
lemma first_countable_neighbourhood_base:
"first_countable X \<longleftrightarrow>
(\<forall>x \<in> topspace X. \<exists>\<B>. countable \<B> \<and> neighbourhood_base_at x (\<lambda>V. V \<in> \<B>) X)" (is "?lhs=?rhs")
proof
assume ?lhs
then show ?rhs
by (metis first_countable_neighbourhood_base_alt)
next
assume R: ?rhs
show ?lhs
unfolding first_countable_neighbourhood_base_alt
proof
fix x
assume "x \<in> topspace X"
with R obtain \<B> where "countable \<B>" and \<B>: "neighbourhood_base_at x (\<lambda>V. V \<in> \<B>) X"
by blast
then
show "\<exists>\<B>. countable \<B> \<and> Ball \<B> (openin X) \<and> neighbourhood_base_at x (\<lambda>V. V \<in> \<B>) X"
unfolding neighbourhood_base_at_def
apply (rule_tac x="(\<lambda>u. X interior_of u) ` \<B>" in exI)
by (smt (verit, best) countable_image image_iff interior_of_eq interior_of_mono openin_interior_of)
qed
qed
subsection\<open>$T_0$ spaces and the Kolmogorov quotient\<close>
definition t0_space where
"t0_space X \<equiv>
\<forall>x \<in> topspace X. \<forall>y \<in> topspace X. x \<noteq> y \<longrightarrow> (\<exists>U. openin X U \<and> (x \<notin> U \<longleftrightarrow> y \<in> U))"
lemma t0_space_expansive:
"\<lbrakk>topspace Y = topspace X; \<And>U. openin X U \<Longrightarrow> openin Y U\<rbrakk> \<Longrightarrow> t0_space X \<Longrightarrow> t0_space Y"
by (metis t0_space_def)
lemma t1_imp_t0_space: "t1_space X \<Longrightarrow> t0_space X"
by (metis t0_space_def t1_space_def)
lemma t1_eq_symmetric_t0_space_alt:
"t1_space X \<longleftrightarrow>
t0_space X \<and>
(\<forall>x \<in> topspace X. \<forall>y \<in> topspace X. x \<in> X closure_of {y} \<longleftrightarrow> y \<in> X closure_of {x})"
apply (simp add: t0_space_def t1_space_def closure_of_def)
by (smt (verit, best) openin_topspace)
lemma t1_eq_symmetric_t0_space:
"t1_space X \<longleftrightarrow> t0_space X \<and> (\<forall>x y. x \<in> X closure_of {y} \<longleftrightarrow> y \<in> X closure_of {x})"
by (auto simp: t1_eq_symmetric_t0_space_alt in_closure_of)
lemma Hausdorff_imp_t0_space:
"Hausdorff_space X \<Longrightarrow> t0_space X"
by (simp add: Hausdorff_imp_t1_space t1_imp_t0_space)
lemma t0_space:
"t0_space X \<longleftrightarrow>
(\<forall>x \<in> topspace X. \<forall>y \<in> topspace X. x \<noteq> y \<longrightarrow> (\<exists>C. closedin X C \<and> (x \<notin> C \<longleftrightarrow> y \<in> C)))"
unfolding t0_space_def by (metis Diff_iff closedin_def openin_closedin_eq)
lemma homeomorphic_t0_space:
assumes "X homeomorphic_space Y"
shows "t0_space X \<longleftrightarrow> t0_space Y"
proof -
obtain f where f: "homeomorphic_map X Y f" and F: "inj_on f (topspace X)" and "topspace Y = f ` topspace X"
by (metis assms homeomorphic_imp_injective_map homeomorphic_imp_surjective_map homeomorphic_space)
with inj_on_image_mem_iff [OF F]
show ?thesis
apply (simp add: t0_space_def homeomorphic_eq_everything_map continuous_map_def open_map_def inj_on_def)
by (smt (verit) mem_Collect_eq openin_subset)
qed
lemma t0_space_closure_of_sing:
"t0_space X \<longleftrightarrow>
(\<forall>x \<in> topspace X. \<forall>y \<in> topspace X. X closure_of {x} = X closure_of {y} \<longrightarrow> x = y)"
by (simp add: t0_space_def closure_of_def set_eq_iff) (smt (verit))
lemma t0_space_discrete_topology: "t0_space (discrete_topology S)"
by (simp add: Hausdorff_imp_t0_space)
lemma t0_space_subtopology: "t0_space X \<Longrightarrow> t0_space (subtopology X U)"
by (simp add: t0_space_def openin_subtopology) (metis Int_iff)
lemma t0_space_retraction_map_image:
"\<lbrakk>retraction_map X Y r; t0_space X\<rbrakk> \<Longrightarrow> t0_space Y"
using hereditary_imp_retractive_property homeomorphic_t0_space t0_space_subtopology by blast
lemma t0_space_prod_topologyI: "\<lbrakk>t0_space X; t0_space Y\<rbrakk> \<Longrightarrow> t0_space (prod_topology X Y)"
by (simp add: t0_space_closure_of_sing closure_of_Times closure_of_eq_empty_gen times_eq_iff flip: sing_Times_sing insert_Times_insert)
lemma t0_space_prod_topology_iff:
"t0_space (prod_topology X Y) \<longleftrightarrow> prod_topology X Y = trivial_topology \<or> t0_space X \<and> t0_space Y" (is "?lhs=?rhs")
proof
assume ?lhs
then show ?rhs
by (metis prod_topology_trivial_iff retraction_map_fst retraction_map_snd t0_space_retraction_map_image)
qed (metis t0_space_discrete_topology t0_space_prod_topologyI)
proposition t0_space_product_topology:
"t0_space (product_topology X I) \<longleftrightarrow> product_topology X I = trivial_topology \<or> (\<forall>i \<in> I. t0_space (X i))"
(is "?lhs=?rhs")
proof
assume ?lhs
then show ?rhs
by (meson retraction_map_product_projection t0_space_retraction_map_image)
next
assume R: ?rhs
show ?lhs
proof (cases "product_topology X I = trivial_topology")
case True
then show ?thesis
by (simp add: t0_space_def)
next
case False
show ?thesis
unfolding t0_space
proof (intro strip)
fix x y
assume x: "x \<in> topspace (product_topology X I)"
and y: "y \<in> topspace (product_topology X I)"
and "x \<noteq> y"
then obtain i where "i \<in> I" "x i \<noteq> y i"
by (metis PiE_ext topspace_product_topology)
then have "t0_space (X i)"
using False R by blast
then obtain U where "closedin (X i) U" "(x i \<notin> U \<longleftrightarrow> y i \<in> U)"
by (metis t0_space PiE_mem \<open>i \<in> I\<close> \<open>x i \<noteq> y i\<close> topspace_product_topology x y)
with \<open>i \<in> I\<close> x y show "\<exists>U. closedin (product_topology X I) U \<and> (x \<notin> U) = (y \<in> U)"
by (rule_tac x="PiE I (\<lambda>j. if j = i then U else topspace(X j))" in exI)
(simp add: closedin_product_topology PiE_iff)
qed
qed
qed
subsection \<open>Kolmogorov quotients\<close>
definition Kolmogorov_quotient
where "Kolmogorov_quotient X \<equiv> \<lambda>x. @y. \<forall>U. openin X U \<longrightarrow> (y \<in> U \<longleftrightarrow> x \<in> U)"
lemma Kolmogorov_quotient_in_open:
"openin X U \<Longrightarrow> (Kolmogorov_quotient X x \<in> U \<longleftrightarrow> x \<in> U)"
by (smt (verit, ccfv_SIG) Kolmogorov_quotient_def someI_ex)
lemma Kolmogorov_quotient_in_topspace:
"Kolmogorov_quotient X x \<in> topspace X \<longleftrightarrow> x \<in> topspace X"
by (simp add: Kolmogorov_quotient_in_open)
lemma Kolmogorov_quotient_in_closed:
"closedin X C \<Longrightarrow> (Kolmogorov_quotient X x \<in> C \<longleftrightarrow> x \<in> C)"
unfolding closedin_def
by (meson DiffD2 DiffI Kolmogorov_quotient_in_open Kolmogorov_quotient_in_topspace in_mono)
lemma continuous_map_Kolmogorov_quotient:
"continuous_map X X (Kolmogorov_quotient X)"
using Kolmogorov_quotient_in_open openin_subopen openin_subset
by (fastforce simp: continuous_map_def Kolmogorov_quotient_in_topspace)
lemma open_map_Kolmogorov_quotient_explicit:
"openin X U \<Longrightarrow> Kolmogorov_quotient X ` U = Kolmogorov_quotient X ` topspace X \<inter> U"
using Kolmogorov_quotient_in_open openin_subset by fastforce
lemma open_map_Kolmogorov_quotient_gen:
"open_map (subtopology X S) (subtopology X (Kolmogorov_quotient X ` S)) (Kolmogorov_quotient X)"
proof (clarsimp simp: open_map_def openin_subtopology_alt image_iff)
fix U
assume "openin X U"
then have "Kolmogorov_quotient X ` (S \<inter> U) = Kolmogorov_quotient X ` S \<inter> U"
using Kolmogorov_quotient_in_open [of X U] by auto
then show "\<exists>V. openin X V \<and> Kolmogorov_quotient X ` (S \<inter> U) = Kolmogorov_quotient X ` S \<inter> V"
using \<open>openin X U\<close> by blast
qed
lemma open_map_Kolmogorov_quotient:
"open_map X (subtopology X (Kolmogorov_quotient X ` topspace X))
(Kolmogorov_quotient X)"
by (metis open_map_Kolmogorov_quotient_gen subtopology_topspace)
lemma closed_map_Kolmogorov_quotient_explicit:
"closedin X U \<Longrightarrow> Kolmogorov_quotient X ` U = Kolmogorov_quotient X ` topspace X \<inter> U"
using closedin_subset by (fastforce simp: Kolmogorov_quotient_in_closed)
lemma closed_map_Kolmogorov_quotient_gen:
"closed_map (subtopology X S) (subtopology X (Kolmogorov_quotient X ` S))
(Kolmogorov_quotient X)"
using Kolmogorov_quotient_in_closed by (force simp: closed_map_def closedin_subtopology_alt image_iff)
lemma closed_map_Kolmogorov_quotient:
"closed_map X (subtopology X (Kolmogorov_quotient X ` topspace X))
(Kolmogorov_quotient X)"
by (metis closed_map_Kolmogorov_quotient_gen subtopology_topspace)
lemma quotient_map_Kolmogorov_quotient_gen:
"quotient_map (subtopology X S) (subtopology X (Kolmogorov_quotient X ` S)) (Kolmogorov_quotient X)"
proof (intro continuous_open_imp_quotient_map)
show "continuous_map (subtopology X S) (subtopology X (Kolmogorov_quotient X ` S)) (Kolmogorov_quotient X)"
by (simp add: continuous_map_Kolmogorov_quotient continuous_map_from_subtopology continuous_map_in_subtopology image_mono)
show "open_map (subtopology X S) (subtopology X (Kolmogorov_quotient X ` S)) (Kolmogorov_quotient X)"
using open_map_Kolmogorov_quotient_gen by blast
show "Kolmogorov_quotient X ` topspace (subtopology X S) = topspace (subtopology X (Kolmogorov_quotient X ` S))"
by (force simp: Kolmogorov_quotient_in_open)
qed
lemma quotient_map_Kolmogorov_quotient:
"quotient_map X (subtopology X (Kolmogorov_quotient X ` topspace X)) (Kolmogorov_quotient X)"
by (metis quotient_map_Kolmogorov_quotient_gen subtopology_topspace)
lemma Kolmogorov_quotient_eq:
"Kolmogorov_quotient X x = Kolmogorov_quotient X y \<longleftrightarrow>
(\<forall>U. openin X U \<longrightarrow> (x \<in> U \<longleftrightarrow> y \<in> U))" (is "?lhs=?rhs")
proof
assume ?lhs then show ?rhs
by (metis Kolmogorov_quotient_in_open)
next
assume ?rhs then show ?lhs
by (simp add: Kolmogorov_quotient_def)
qed
lemma Kolmogorov_quotient_eq_alt:
"Kolmogorov_quotient X x = Kolmogorov_quotient X y \<longleftrightarrow>
(\<forall>U. closedin X U \<longrightarrow> (x \<in> U \<longleftrightarrow> y \<in> U))" (is "?lhs=?rhs")
proof
assume ?lhs then show ?rhs
by (metis Kolmogorov_quotient_in_closed)
next
assume ?rhs then show ?lhs
by (smt (verit) Diff_iff Kolmogorov_quotient_eq closedin_topspace in_mono openin_closedin_eq)
qed
lemma Kolmogorov_quotient_continuous_map:
assumes "continuous_map X Y f" "t0_space Y" and x: "x \<in> topspace X"
shows "f (Kolmogorov_quotient X x) = f x"
using assms unfolding continuous_map_def t0_space_def
by (smt (verit, ccfv_threshold) Kolmogorov_quotient_in_open Kolmogorov_quotient_in_topspace PiE mem_Collect_eq)
lemma t0_space_Kolmogorov_quotient:
"t0_space (subtopology X (Kolmogorov_quotient X ` topspace X))"
apply (clarsimp simp: t0_space_def )
by (smt (verit, best) Kolmogorov_quotient_eq imageE image_eqI open_map_Kolmogorov_quotient open_map_def)
lemma Kolmogorov_quotient_id:
"t0_space X \<Longrightarrow> x \<in> topspace X \<Longrightarrow> Kolmogorov_quotient X x = x"
by (metis Kolmogorov_quotient_in_open Kolmogorov_quotient_in_topspace t0_space_def)
lemma Kolmogorov_quotient_idemp:
"Kolmogorov_quotient X (Kolmogorov_quotient X x) = Kolmogorov_quotient X x"
by (simp add: Kolmogorov_quotient_eq Kolmogorov_quotient_in_open)
lemma retraction_maps_Kolmogorov_quotient:
"retraction_maps X
(subtopology X (Kolmogorov_quotient X ` topspace X))
(Kolmogorov_quotient X) id"
unfolding retraction_maps_def continuous_map_in_subtopology
using Kolmogorov_quotient_idemp continuous_map_Kolmogorov_quotient by force
lemma retraction_map_Kolmogorov_quotient:
"retraction_map X
(subtopology X (Kolmogorov_quotient X ` topspace X))
(Kolmogorov_quotient X)"
using retraction_map_def retraction_maps_Kolmogorov_quotient by blast
lemma retract_of_space_Kolmogorov_quotient_image:
"Kolmogorov_quotient X ` topspace X retract_of_space X"
proof -
have "continuous_map X X (Kolmogorov_quotient X)"
by (simp add: continuous_map_Kolmogorov_quotient)
then have "Kolmogorov_quotient X ` topspace X \<subseteq> topspace X"
by (simp add: continuous_map_image_subset_topspace)
then show ?thesis
by (meson retract_of_space_retraction_maps retraction_maps_Kolmogorov_quotient)
qed
lemma Kolmogorov_quotient_lift_exists:
assumes "S \<subseteq> topspace X" "t0_space Y" and f: "continuous_map (subtopology X S) Y f"
obtains g where "continuous_map (subtopology X (Kolmogorov_quotient X ` S)) Y g"
"\<And>x. x \<in> S \<Longrightarrow> g(Kolmogorov_quotient X x) = f x"
proof -
have "\<And>x y. \<lbrakk>x \<in> S; y \<in> S; Kolmogorov_quotient X x = Kolmogorov_quotient X y\<rbrakk> \<Longrightarrow> f x = f y"
using assms
apply (simp add: Kolmogorov_quotient_eq t0_space_def continuous_map_def Int_absorb1 openin_subtopology)
by (smt (verit, del_insts) Int_iff mem_Collect_eq Pi_iff)
then obtain g where g: "continuous_map (subtopology X (Kolmogorov_quotient X ` S)) Y g"
"g ` (topspace X \<inter> Kolmogorov_quotient X ` S) = f ` S"
"\<And>x. x \<in> S \<Longrightarrow> g (Kolmogorov_quotient X x) = f x"
using quotient_map_lift_exists [OF quotient_map_Kolmogorov_quotient_gen [of X S] f]
by (metis assms(1) topspace_subtopology topspace_subtopology_subset)
show ?thesis
proof qed (use g in auto)
qed
subsection\<open>Closed diagonals and graphs\<close>
lemma Hausdorff_space_closedin_diagonal:
"Hausdorff_space X \<longleftrightarrow> closedin (prod_topology X X) ((\<lambda>x. (x,x)) ` topspace X)"
proof -
have \<section>: "((\<lambda>x. (x, x)) ` topspace X) \<subseteq> topspace X \<times> topspace X"
by auto
show ?thesis
apply (simp add: closedin_def openin_prod_topology_alt Hausdorff_space_def disjnt_iff \<section>)
apply (intro all_cong1 imp_cong ex_cong1 conj_cong refl)
by (force dest!: openin_subset)+
qed
lemma closed_map_diag_eq:
"closed_map X (prod_topology X X) (\<lambda>x. (x,x)) \<longleftrightarrow> Hausdorff_space X"
proof -
have "section_map X (prod_topology X X) (\<lambda>x. (x, x))"
unfolding section_map_def retraction_maps_def
by (smt (verit) continuous_map_fst continuous_map_of_fst continuous_map_on_empty continuous_map_pairwise fst_conv fst_diag_fst snd_diag_fst)
then have "embedding_map X (prod_topology X X) (\<lambda>x. (x, x))"
by (rule section_imp_embedding_map)
then show ?thesis
using Hausdorff_space_closedin_diagonal embedding_imp_closed_map_eq by blast
qed
lemma proper_map_diag_eq [simp]:
"proper_map X (prod_topology X X) (\<lambda>x. (x,x)) \<longleftrightarrow> Hausdorff_space X"
by (simp add: closed_map_diag_eq inj_on_convol_ident injective_imp_proper_eq_closed_map)
lemma closedin_continuous_maps_eq:
assumes "Hausdorff_space Y" and f: "continuous_map X Y f" and g: "continuous_map X Y g"
shows "closedin X {x \<in> topspace X. f x = g x}"
proof -
have \<section>:"{x \<in> topspace X. f x = g x} = {x \<in> topspace X. (f x,g x) \<in> ((\<lambda>y.(y,y)) ` topspace Y)}"
using f continuous_map_image_subset_topspace by fastforce
show ?thesis
unfolding \<section>
proof (intro closedin_continuous_map_preimage)
show "continuous_map X (prod_topology Y Y) (\<lambda>x. (f x, g x))"
by (simp add: continuous_map_pairedI f g)
show "closedin (prod_topology Y Y) ((\<lambda>y. (y, y)) ` topspace Y)"
using Hausdorff_space_closedin_diagonal assms by blast
qed
qed
lemma forall_in_closure_of:
assumes "x \<in> X closure_of S" "\<And>x. x \<in> S \<Longrightarrow> P x"
and "closedin X {x \<in> topspace X. P x}"
shows "P x"
by (smt (verit, ccfv_threshold) Diff_iff assms closedin_def in_closure_of mem_Collect_eq)
lemma forall_in_closure_of_eq:
assumes x: "x \<in> X closure_of S"
and Y: "Hausdorff_space Y"
and f: "continuous_map X Y f" and g: "continuous_map X Y g"
and fg: "\<And>x. x \<in> S \<Longrightarrow> f x = g x"
shows "f x = g x"
proof -
have "closedin X {x \<in> topspace X. f x = g x}"
using Y closedin_continuous_maps_eq f g by blast
then show ?thesis
using forall_in_closure_of [OF x fg]
by fastforce
qed
lemma retract_of_space_imp_closedin:
assumes "Hausdorff_space X" and S: "S retract_of_space X"
shows "closedin X S"
proof -
obtain r where r: "continuous_map X (subtopology X S) r" "\<forall>x\<in>S. r x = x"
using assms by (meson retract_of_space_def)
then have \<section>: "S = {x \<in> topspace X. r x = x}"
using S retract_of_space_imp_subset by (force simp: continuous_map_def Pi_iff)
show ?thesis
unfolding \<section>
using r continuous_map_into_fulltopology assms
by (force intro: closedin_continuous_maps_eq)
qed
lemma homeomorphic_maps_graph:
"homeomorphic_maps X (subtopology (prod_topology X Y) ((\<lambda>x. (x, f x)) ` (topspace X)))
(\<lambda>x. (x, f x)) fst \<longleftrightarrow> continuous_map X Y f"
(is "?lhs=?rhs")
proof
assume ?lhs
then
have h: "homeomorphic_map X (subtopology (prod_topology X Y) ((\<lambda>x. (x, f x)) ` topspace X)) (\<lambda>x. (x, f x))"
by (auto simp: homeomorphic_maps_map)
have "f = snd \<circ> (\<lambda>x. (x, f x))"
by force
then show ?rhs
by (metis (no_types, lifting) h continuous_map_in_subtopology continuous_map_snd_of homeomorphic_eq_everything_map)
next
assume ?rhs
then show ?lhs
unfolding homeomorphic_maps_def
by (smt (verit, del_insts) continuous_map_eq continuous_map_subtopology_fst embedding_map_def
embedding_map_graph homeomorphic_eq_everything_map image_cong image_iff prod.sel(1))
qed
subsection \<open> KC spaces, those where all compact sets are closed.\<close>
definition kc_space
where "kc_space X \<equiv> \<forall>S. compactin X S \<longrightarrow> closedin X S"
lemma kc_space_euclidean: "kc_space (euclidean :: 'a::metric_space topology)"
by (simp add: compact_imp_closed kc_space_def)
lemma kc_space_expansive:
"\<lbrakk>kc_space X; topspace Y = topspace X; \<And>U. openin X U \<Longrightarrow> openin Y U\<rbrakk>
\<Longrightarrow> kc_space Y"
by (meson compactin_contractive kc_space_def topology_finer_closedin)
lemma compactin_imp_closedin_gen:
"\<lbrakk>kc_space X; compactin X S\<rbrakk> \<Longrightarrow> closedin X S"
using kc_space_def by blast
lemma Hausdorff_imp_kc_space: "Hausdorff_space X \<Longrightarrow> kc_space X"
by (simp add: compactin_imp_closedin kc_space_def)
lemma kc_imp_t1_space: "kc_space X \<Longrightarrow> t1_space X"
by (simp add: finite_imp_compactin kc_space_def t1_space_closedin_finite)
lemma kc_space_subtopology:
"kc_space X \<Longrightarrow> kc_space(subtopology X S)"
by (metis closedin_Int_closure_of closure_of_eq compactin_subtopology inf.absorb2 kc_space_def)
lemma kc_space_discrete_topology:
"kc_space(discrete_topology U)"
using Hausdorff_space_discrete_topology compactin_imp_closedin kc_space_def by blast
lemma kc_space_continuous_injective_map_preimage:
assumes "kc_space Y" "continuous_map X Y f" and injf: "inj_on f (topspace X)"
shows "kc_space X"
unfolding kc_space_def
proof (intro strip)
fix S
assume S: "compactin X S"
have "S = {x \<in> topspace X. f x \<in> f ` S}"
using S compactin_subset_topspace inj_onD [OF injf] by blast
with assms S show "closedin X S"
by (metis (no_types, lifting) Collect_cong closedin_continuous_map_preimage compactin_imp_closedin_gen image_compactin)
qed
lemma kc_space_retraction_map_image:
assumes "retraction_map X Y r" "kc_space X"
shows "kc_space Y"
proof -
obtain s where s: "continuous_map X Y r" "continuous_map Y X s" "\<And>x. x \<in> topspace Y \<Longrightarrow> r (s x) = x"
using assms by (force simp: retraction_map_def retraction_maps_def)
then have inj: "inj_on s (topspace Y)"
by (metis inj_on_def)
show ?thesis
unfolding kc_space_def
proof (intro strip)
fix S
assume S: "compactin Y S"
have "S = {x \<in> topspace Y. s x \<in> s ` S}"
using S compactin_subset_topspace inj_onD [OF inj] by blast
with assms S show "closedin Y S"
by (meson compactin_imp_closedin_gen inj kc_space_continuous_injective_map_preimage s(2))
qed
qed
lemma homeomorphic_kc_space:
"X homeomorphic_space Y \<Longrightarrow> kc_space X \<longleftrightarrow> kc_space Y"
by (meson homeomorphic_eq_everything_map homeomorphic_space homeomorphic_space_sym kc_space_continuous_injective_map_preimage)
lemma compact_kc_eq_maximal_compact_space:
assumes "compact_space X"
shows "kc_space X \<longleftrightarrow>
(\<forall>Y. topspace Y = topspace X \<and> (\<forall>S. openin X S \<longrightarrow> openin Y S) \<and> compact_space Y \<longrightarrow> Y = X)" (is "?lhs=?rhs")
proof
assume ?lhs
then show ?rhs
by (metis closedin_compact_space compactin_contractive kc_space_def topology_eq topology_finer_closedin)
next
assume R: ?rhs
show ?lhs
unfolding kc_space_def
proof (intro strip)
fix S
assume S: "compactin X S"
define Y where
"Y \<equiv> topology (arbitrary union_of (finite intersection_of (\<lambda>A. A = topspace X - S \<or> openin X A)
relative_to (topspace X)))"
have "topspace Y = topspace X"
by (auto simp: Y_def)
have "openin X T \<longrightarrow> openin Y T" for T
by (simp add: Y_def arbitrary_union_of_inc finite_intersection_of_inc openin_subbase openin_subset relative_to_subset_inc)
have "compact_space Y"
proof (rule Alexander_subbase_alt)
show "\<exists>\<F>'. finite \<F>' \<and> \<F>' \<subseteq> \<C> \<and> topspace X \<subseteq> \<Union> \<F>'"
if \<C>: "\<C> \<subseteq> insert (topspace X - S) (Collect (openin X))" and sub: "topspace X \<subseteq> \<Union>\<C>" for \<C>
proof -
consider "\<C> \<subseteq> Collect (openin X)" | \<V> where "\<C> = insert (topspace X - S) \<V>" "\<V> \<subseteq> Collect (openin X)"
using \<C> by (metis insert_Diff subset_insert_iff)
then show ?thesis
proof cases
case 1
then show ?thesis
by (metis assms compact_space_alt mem_Collect_eq subsetD that(2))
next
case 2
then have "S \<subseteq> \<Union>\<V>"
using S sub compactin_subset_topspace by blast
with 2 obtain \<F> where "finite \<F> \<and> \<F> \<subseteq> \<V> \<and> S \<subseteq> \<Union>\<F>"
using S unfolding compactin_def by (metis Ball_Collect)
with 2 show ?thesis
by (rule_tac x="insert (topspace X - S) \<F>" in exI) blast
qed
qed
qed (auto simp: Y_def)
have "Y = X"
using R \<open>\<And>S. openin X S \<longrightarrow> openin Y S\<close> \<open>compact_space Y\<close> \<open>topspace Y = topspace X\<close> by blast
moreover have "openin Y (topspace X - S)"
by (simp add: Y_def arbitrary_union_of_inc finite_intersection_of_inc openin_subbase relative_to_subset_inc)
ultimately show "closedin X S"
unfolding closedin_def using S compactin_subset_topspace by blast
qed
qed
lemma continuous_imp_closed_map_gen:
"\<lbrakk>compact_space X; kc_space Y; continuous_map X Y f\<rbrakk> \<Longrightarrow> closed_map X Y f"
by (meson closed_map_def closedin_compact_space compactin_imp_closedin_gen image_compactin)
lemma kc_space_compact_subtopologies:
"kc_space X \<longleftrightarrow> (\<forall>K. compactin X K \<longrightarrow> kc_space(subtopology X K))" (is "?lhs=?rhs")
proof
assume ?lhs
then show ?rhs
by (auto simp: kc_space_def closedin_closed_subtopology compactin_subtopology)
next
assume R: ?rhs
show ?lhs
unfolding kc_space_def
proof (intro strip)
fix K
assume K: "compactin X K"
then have "K \<subseteq> topspace X"
by (simp add: compactin_subset_topspace)
moreover have "X closure_of K \<subseteq> K"
proof
fix x
assume x: "x \<in> X closure_of K"
have "kc_space (subtopology X K)"
by (simp add: R \<open>compactin X K\<close>)
have "compactin X (insert x K)"
by (metis K x compactin_Un compactin_sing in_closure_of insert_is_Un)
then show "x \<in> K"
by (metis R x K Int_insert_left_if1 closedin_Int_closure_of compact_imp_compactin_subtopology
insertCI kc_space_def subset_insertI)
qed
ultimately show "closedin X K"
using closure_of_subset_eq by blast
qed
qed
lemma kc_space_compact_prod_topology:
assumes "compact_space X"
shows "kc_space(prod_topology X X) \<longleftrightarrow> Hausdorff_space X" (is "?lhs=?rhs")
proof
assume L: ?lhs
show ?rhs
unfolding closed_map_diag_eq [symmetric]
proof (intro continuous_imp_closed_map_gen)
show "continuous_map X (prod_topology X X) (\<lambda>x. (x, x))"
by (intro continuous_intros)
qed (use L assms in auto)
next
assume ?rhs then show ?lhs
by (simp add: Hausdorff_imp_kc_space Hausdorff_space_prod_topology)
qed
lemma kc_space_prod_topology:
"kc_space(prod_topology X X) \<longleftrightarrow> (\<forall>K. compactin X K \<longrightarrow> Hausdorff_space(subtopology X K))" (is "?lhs=?rhs")
proof
assume ?lhs
then show ?rhs
by (metis compactin_subspace kc_space_compact_prod_topology kc_space_subtopology subtopology_Times)
next
assume R: ?rhs
have "kc_space (subtopology (prod_topology X X) L)" if "compactin (prod_topology X X) L" for L
proof -
define K where "K \<equiv> fst ` L \<union> snd ` L"
have "L \<subseteq> K \<times> K"
by (force simp: K_def)
have "compactin X K"
by (metis K_def compactin_Un continuous_map_fst continuous_map_snd image_compactin that)
then have "Hausdorff_space (subtopology X K)"
by (simp add: R)
then have "kc_space (prod_topology (subtopology X K) (subtopology X K))"
by (simp add: \<open>compactin X K\<close> compact_space_subtopology kc_space_compact_prod_topology)
then have "kc_space (subtopology (prod_topology (subtopology X K) (subtopology X K)) L)"
using kc_space_subtopology by blast
then show ?thesis
using \<open>L \<subseteq> K \<times> K\<close> subtopology_Times subtopology_subtopology
by (metis (no_types, lifting) Sigma_cong inf.absorb_iff2)
qed
then show ?lhs
using kc_space_compact_subtopologies by blast
qed
lemma kc_space_prod_topology_alt:
"kc_space(prod_topology X X) \<longleftrightarrow>
kc_space X \<and>
(\<forall>K. compactin X K \<longrightarrow> Hausdorff_space(subtopology X K))"
using Hausdorff_imp_kc_space kc_space_compact_subtopologies kc_space_prod_topology by blast
proposition kc_space_prod_topology_left:
assumes X: "kc_space X" and Y: "Hausdorff_space Y"
shows "kc_space (prod_topology X Y)"
unfolding kc_space_def
proof (intro strip)
fix K
assume K: "compactin (prod_topology X Y) K"
then have "K \<subseteq> topspace X \<times> topspace Y"
using compactin_subset_topspace topspace_prod_topology by blast
moreover have "\<exists>T. openin (prod_topology X Y) T \<and> (a,b) \<in> T \<and> T \<subseteq> (topspace X \<times> topspace Y) - K"
if ab: "(a,b) \<in> (topspace X \<times> topspace Y) - K" for a b
proof -
have "compactin Y {b}"
using that by force
moreover
have "compactin Y {y \<in> topspace Y. (a,y) \<in> K}"
proof -
have "compactin (prod_topology X Y) (K \<inter> {a} \<times> topspace Y)"
using that compact_Int_closedin [OF K]
by (simp add: X closedin_prod_Times_iff compactin_imp_closedin_gen)
moreover have "subtopology (prod_topology X Y) (K \<inter> {a} \<times> topspace Y) homeomorphic_space
subtopology Y {y \<in> topspace Y. (a, y) \<in> K}"
unfolding homeomorphic_space_def homeomorphic_maps_def
using that
apply (rule_tac x="snd" in exI)
apply (rule_tac x="Pair a" in exI)
by (force simp: continuous_map_in_subtopology continuous_map_from_subtopology continuous_map_subtopology_snd continuous_map_paired)
ultimately show ?thesis
by (simp add: compactin_subspace homeomorphic_compact_space)
qed
moreover have "disjnt {b} {y \<in> topspace Y. (a,y) \<in> K}"
using ab by force
ultimately obtain V U where VU: "openin Y V" "openin Y U" "{b} \<subseteq> V" "{y \<in> topspace Y. (a,y) \<in> K} \<subseteq> U" "disjnt V U"
using Hausdorff_space_compact_separation [OF Y] by blast
define V' where "V' \<equiv> topspace Y - U"
have W: "closedin Y V'" "{y \<in> topspace Y. (a,y) \<in> K} \<subseteq> topspace Y - V'" "disjnt V (topspace Y - V')"
using VU by (auto simp: V'_def disjnt_iff)
with VU obtain "V \<subseteq> topspace Y" "V' \<subseteq> topspace Y"
by (meson closedin_subset openin_closedin_eq)
then obtain "b \<in> V" "disjnt {y \<in> topspace Y. (a,y) \<in> K} V'" "V \<subseteq> V'"
using VU unfolding disjnt_iff V'_def by force
define C where "C \<equiv> image fst (K \<inter> {z \<in> topspace(prod_topology X Y). snd z \<in> V'})"
have "closedin (prod_topology X Y) {z \<in> topspace (prod_topology X Y). snd z \<in> V'}"
using closedin_continuous_map_preimage \<open>closedin Y V'\<close> continuous_map_snd by blast
then have "compactin X C"
unfolding C_def by (meson K compact_Int_closedin continuous_map_fst image_compactin)
then have "closedin X C"
using assms by (auto simp: kc_space_def)
show ?thesis
proof (intro exI conjI)
show "openin (prod_topology X Y) ((topspace X - C) \<times> V)"
by (simp add: VU \<open>closedin X C\<close> openin_diff openin_prod_Times_iff)
have "a \<notin> C"
using VU by (auto simp: C_def V'_def)
then show "(a, b) \<in> (topspace X - C) \<times> V"
using \<open>a \<notin> C\<close> \<open>b \<in> V\<close> that by blast
show "(topspace X - C) \<times> V \<subseteq> topspace X \<times> topspace Y - K"
using \<open>V \<subseteq> V'\<close> \<open>V \<subseteq> topspace Y\<close>
apply (simp add: C_def )
by (smt (verit, ccfv_threshold) DiffE DiffI IntI SigmaE SigmaI image_eqI mem_Collect_eq prod.sel(1) snd_conv subset_iff)
qed
qed
ultimately show "closedin (prod_topology X Y) K"
by (metis surj_pair closedin_def openin_subopen topspace_prod_topology)
qed
lemma kc_space_prod_topology_right:
"\<lbrakk>Hausdorff_space X; kc_space Y\<rbrakk> \<Longrightarrow> kc_space (prod_topology X Y)"
using kc_space_prod_topology_left homeomorphic_kc_space homeomorphic_space_prod_topology_swap by blast
subsection \<open>Technical results about proper maps, perfect maps, etc\<close>
lemma compact_imp_proper_map_gen:
assumes Y: "\<And>S. \<lbrakk>S \<subseteq> topspace Y; \<And>K. compactin Y K \<Longrightarrow> compactin Y (S \<inter> K)\<rbrakk>
\<Longrightarrow> closedin Y S"
and fim: "f ` (topspace X) \<subseteq> topspace Y"
and f: "continuous_map X Y f \<or> kc_space X"
and YX: "\<And>K. compactin Y K \<Longrightarrow> compactin X {x \<in> topspace X. f x \<in> K}"
shows "proper_map X Y f"
unfolding proper_map_alt closed_map_def
proof (intro conjI strip)
fix C
assume C: "closedin X C"
show "closedin Y (f ` C)"
proof (intro Y)
show "f ` C \<subseteq> topspace Y"
using C closedin_subset fim by blast
fix K
assume K: "compactin Y K"
define A where "A \<equiv> {x \<in> topspace X. f x \<in> K}"
have eq: "f ` C \<inter> K = f ` ({x \<in> topspace X. f x \<in> K} \<inter> C)"
using C closedin_subset by auto
show "compactin Y (f ` C \<inter> K)"
unfolding eq
proof (rule image_compactin)
show "compactin (subtopology X A) ({x \<in> topspace X. f x \<in> K} \<inter> C)"
proof (rule closedin_compact_space)
show "compact_space (subtopology X A)"
by (simp add: A_def K YX compact_space_subtopology)
show "closedin (subtopology X A) ({x \<in> topspace X. f x \<in> K} \<inter> C)"
using A_def C closedin_subtopology by blast
qed
have "continuous_map (subtopology X A) (subtopology Y K) f" if "kc_space X"
unfolding continuous_map_closedin
proof (intro conjI strip)
show "f \<in> topspace (subtopology X A) \<rightarrow> topspace (subtopology Y K)"
using A_def K compactin_subset_topspace by fastforce
next
fix C
assume C: "closedin (subtopology Y K) C"
show "closedin (subtopology X A) {x \<in> topspace (subtopology X A). f x \<in> C}"
proof (rule compactin_imp_closedin_gen)
show "kc_space (subtopology X A)"
by (simp add: kc_space_subtopology that)
have [simp]: "{x \<in> topspace X. f x \<in> K \<and> f x \<in> C} = {x \<in> topspace X. f x \<in> C}"
using C closedin_imp_subset by auto
have "compactin (subtopology Y K) C"
by (simp add: C K closedin_compact_space compact_space_subtopology)
then have "compactin X {x \<in> topspace X. x \<in> A \<and> f x \<in> C}"
by (auto simp: A_def compactin_subtopology dest: YX)
then show "compactin (subtopology X A) {x \<in> topspace (subtopology X A). f x \<in> C}"
by (auto simp add: compactin_subtopology)
qed
qed
with f show "continuous_map (subtopology X A) Y f"
using continuous_map_from_subtopology continuous_map_in_subtopology by blast
qed
qed
qed (simp add: YX)
lemma tube_lemma_left:
assumes W: "openin (prod_topology X Y) W" and C: "compactin X C"
and y: "y \<in> topspace Y" and subW: "C \<times> {y} \<subseteq> W"
shows "\<exists>U V. openin X U \<and> openin Y V \<and> C \<subseteq> U \<and> y \<in> V \<and> U \<times> V \<subseteq> W"
proof (cases "C = {}")
case True
with y show ?thesis by auto
next
case False
have "\<exists>U V. openin X U \<and> openin Y V \<and> x \<in> U \<and> y \<in> V \<and> U \<times> V \<subseteq> W"
if "x \<in> C" for x
using W openin_prod_topology_alt subW subsetD that by fastforce
then obtain U V where UV: "\<And>x. x \<in> C \<Longrightarrow> openin X (U x) \<and> openin Y (V x) \<and> x \<in> U x \<and> y \<in> V x \<and> U x \<times> V x \<subseteq> W"
by metis
then obtain D where D: "finite D" "D \<subseteq> C" "C \<subseteq> \<Union> (U ` D)"
using compactinD [OF C, of "U`C"]
by (smt (verit) UN_I finite_subset_image imageE subsetI)
show ?thesis
proof (intro exI conjI)
show "openin X (\<Union> (U ` D))" "openin Y (\<Inter> (V ` D))"
using D False UV by blast+
show "y \<in> \<Inter> (V ` D)" "C \<subseteq> \<Union> (U ` D)" "\<Union>(U ` D) \<times> \<Inter>(V ` D) \<subseteq> W"
using D UV by force+
qed
qed
lemma Wallace_theorem_prod_topology:
assumes "compactin X K" "compactin Y L"
and W: "openin (prod_topology X Y) W" and subW: "K \<times> L \<subseteq> W"
obtains U V where "openin X U" "openin Y V" "K \<subseteq> U" "L \<subseteq> V" "U \<times> V \<subseteq> W"
proof -
have "\<And>y. y \<in> L \<Longrightarrow> \<exists>U V. openin X U \<and> openin Y V \<and> K \<subseteq> U \<and> y \<in> V \<and> U \<times> V \<subseteq> W"
proof (intro tube_lemma_left assms)
fix y assume "y \<in> L"
show "y \<in> topspace Y"
using assms \<open>y \<in> L\<close> compactin_subset_topspace by blast
show "K \<times> {y} \<subseteq> W"
using \<open>y \<in> L\<close> subW by force
qed
then obtain U V where UV:
"\<And>y. y \<in> L \<Longrightarrow> openin X (U y) \<and> openin Y (V y) \<and> K \<subseteq> U y \<and> y \<in> V y \<and> U y \<times> V y \<subseteq> W"
by metis
then obtain M where "finite M" "M \<subseteq> L" and M: "L \<subseteq> \<Union> (V ` M)"
using \<open>compactin Y L\<close> unfolding compactin_def
by (smt (verit) UN_iff finite_subset_image imageE subset_iff)
show thesis
proof (cases "M={}")
case True
with M have "L={}"
by blast
then show ?thesis
using \<open>compactin X K\<close> compactin_subset_topspace that by fastforce
next
case False
show ?thesis
proof
show "openin X (\<Inter>(U`M))"
using False UV \<open>M \<subseteq> L\<close> \<open>finite M\<close> by blast
show "openin Y (\<Union>(V`M))"
using UV \<open>M \<subseteq> L\<close> by blast
show "K \<subseteq> \<Inter>(U`M)"
by (meson INF_greatest UV \<open>M \<subseteq> L\<close> subsetD)
show "L \<subseteq> \<Union>(V`M)"
by (simp add: M)
show "\<Inter>(U`M) \<times> \<Union>(V`M) \<subseteq> W"
using UV \<open>M \<subseteq> L\<close> by fastforce
qed
qed
qed
lemma proper_map_prod:
"proper_map (prod_topology X Y) (prod_topology X' Y') (\<lambda>(x,y). (f x, g y)) \<longleftrightarrow>
(prod_topology X Y) = trivial_topology \<or> proper_map X X' f \<and> proper_map Y Y' g"
(is "?lhs \<longleftrightarrow> _ \<or> ?rhs")
proof (cases "(prod_topology X Y) = trivial_topology")
case True then show ?thesis by auto
next
case False
then have ne: "topspace X \<noteq> {}" "topspace Y \<noteq> {}"
by auto
define h where "h \<equiv> \<lambda>(x,y). (f x, g y)"
have "proper_map X X' f" "proper_map Y Y' g" if ?lhs
proof -
have cm: "closed_map X X' f" "closed_map Y Y' g"
using that False closed_map_prod proper_imp_closed_map by blast+
show "proper_map X X' f"
proof (clarsimp simp add: proper_map_def cm)
fix y
assume y: "y \<in> topspace X'"
obtain z where z: "z \<in> topspace Y"
using ne by blast
then have eq: "{x \<in> topspace X. f x = y} =
fst ` {u \<in> topspace X \<times> topspace Y. h u = (y,g z)}"
by (force simp: h_def)
show "compactin X {x \<in> topspace X. f x = y}"
unfolding eq
proof (intro image_compactin)
have "g z \<in> topspace Y'"
by (meson closed_map_def closedin_subset closedin_topspace cm image_subset_iff z)
with y show "compactin (prod_topology X Y) {u \<in> topspace X \<times> topspace Y. (h u) = (y, g z)}"
using that by (simp add: h_def proper_map_def)
show "continuous_map (prod_topology X Y) X fst"
by (simp add: continuous_map_fst)
qed
qed
show "proper_map Y Y' g"
proof (clarsimp simp add: proper_map_def cm)
fix y
assume y: "y \<in> topspace Y'"
obtain z where z: "z \<in> topspace X"
using ne by blast
then have eq: "{x \<in> topspace Y. g x = y} =
snd ` {u \<in> topspace X \<times> topspace Y. h u = (f z,y)}"
by (force simp: h_def)
show "compactin Y {x \<in> topspace Y. g x = y}"
unfolding eq
proof (intro image_compactin)
have "f z \<in> topspace X'"
by (meson closed_map_def closedin_subset closedin_topspace cm image_subset_iff z)
with y show "compactin (prod_topology X Y) {u \<in> topspace X \<times> topspace Y. (h u) = (f z, y)}"
using that by (simp add: proper_map_def h_def)
show "continuous_map (prod_topology X Y) Y snd"
by (simp add: continuous_map_snd)
qed
qed
qed
moreover
{ assume R: ?rhs
then have fgim: "f \<in> topspace X \<rightarrow> topspace X'" "g \<in> topspace Y \<rightarrow> topspace Y'"
and cm: "closed_map X X' f" "closed_map Y Y' g"
by (auto simp: proper_map_def closed_map_imp_subset_topspace)
have "closed_map (prod_topology X Y) (prod_topology X' Y') h"
unfolding closed_map_fibre_neighbourhood imp_conjL
proof (intro conjI strip)
show "h \<in> topspace (prod_topology X Y) \<rightarrow> topspace (prod_topology X' Y')"
unfolding h_def using fgim by auto
fix W w
assume W: "openin (prod_topology X Y) W"
and w: "w \<in> topspace (prod_topology X' Y')"
and subW: "{x \<in> topspace (prod_topology X Y). h x = w} \<subseteq> W"
then obtain x' y' where weq: "w = (x',y')" "x' \<in> topspace X'" "y' \<in> topspace Y'"
by auto
have eq: "{u \<in> topspace X \<times> topspace Y. h u = (x',y')} = {x \<in> topspace X. f x = x'} \<times> {y \<in> topspace Y. g y = y'}"
by (auto simp: h_def)
obtain U V where "openin X U" "openin Y V" "U \<times> V \<subseteq> W"
and U: "{x \<in> topspace X. f x = x'} \<subseteq> U"
and V: "{x \<in> topspace Y. g x = y'} \<subseteq> V"
proof (rule Wallace_theorem_prod_topology)
show "compactin X {x \<in> topspace X. f x = x'}" "compactin Y {x \<in> topspace Y. g x = y'}"
using R weq unfolding proper_map_def closed_map_fibre_neighbourhood by fastforce+
show "{x \<in> topspace X. f x = x'} \<times> {x \<in> topspace Y. g x = y'} \<subseteq> W"
using weq subW by (auto simp: h_def)
qed (use W in auto)
obtain U' where "openin X' U'" "x' \<in> U'" and U': "{x \<in> topspace X. f x \<in> U'} \<subseteq> U"
using cm U \<open>openin X U\<close> weq unfolding closed_map_fibre_neighbourhood by meson
obtain V' where "openin Y' V'" "y' \<in> V'" and V': "{x \<in> topspace Y. g x \<in> V'} \<subseteq> V"
using cm V \<open>openin Y V\<close> weq unfolding closed_map_fibre_neighbourhood by meson
show "\<exists>V. openin (prod_topology X' Y') V \<and> w \<in> V \<and> {x \<in> topspace (prod_topology X Y). h x \<in> V} \<subseteq> W"
proof (intro conjI exI)
show "openin (prod_topology X' Y') (U' \<times> V')"
by (simp add: \<open>openin X' U'\<close> \<open>openin Y' V'\<close> openin_prod_Times_iff)
show "w \<in> U' \<times> V'"
using \<open>x' \<in> U'\<close> \<open>y' \<in> V'\<close> weq by blast
show "{x \<in> topspace (prod_topology X Y). h x \<in> U' \<times> V'} \<subseteq> W"
using \<open>U \<times> V \<subseteq> W\<close> U' V' h_def by auto
qed
qed
moreover
have "compactin (prod_topology X Y) {u \<in> topspace X \<times> topspace Y. h u = (w, z)}"
if "w \<in> topspace X'" and "z \<in> topspace Y'" for w z
proof -
have eq: "{u \<in> topspace X \<times> topspace Y. h u = (w,z)} =
{u \<in> topspace X. f u = w} \<times> {y. y \<in> topspace Y \<and> g y = z}"
by (auto simp: h_def)
show ?thesis
using R that by (simp add: eq compactin_Times proper_map_def)
qed
ultimately have ?lhs
by (auto simp: h_def proper_map_def)
}
ultimately show ?thesis using False by metis
qed
lemma proper_map_paired:
assumes "Hausdorff_space X \<and> proper_map X Y f \<and> proper_map X Z g \<or>
Hausdorff_space Y \<and> continuous_map X Y f \<and> proper_map X Z g \<or>
Hausdorff_space Z \<and> proper_map X Y f \<and> continuous_map X Z g"
shows "proper_map X (prod_topology Y Z) (\<lambda>x. (f x,g x))"
using assms
proof (elim disjE conjE)
assume \<section>: "Hausdorff_space X" "proper_map X Y f" "proper_map X Z g"
have eq: "(\<lambda>x. (f x, g x)) = (\<lambda>(x, y). (f x, g y)) \<circ> (\<lambda>x. (x, x))"
by auto
show "proper_map X (prod_topology Y Z) (\<lambda>x. (f x, g x))"
unfolding eq
proof (rule proper_map_compose)
show "proper_map X (prod_topology X X) (\<lambda>x. (x,x))"
by (simp add: \<section>)
show "proper_map (prod_topology X X) (prod_topology Y Z) (\<lambda>(x,y). (f x, g y))"
by (simp add: \<section> proper_map_prod)
qed
next
assume \<section>: "Hausdorff_space Y" "continuous_map X Y f" "proper_map X Z g"
have eq: "(\<lambda>x. (f x, g x)) = (\<lambda>(x,y). (x,g y)) \<circ> (\<lambda>x. (f x,x))"
by auto
show "proper_map X (prod_topology Y Z) (\<lambda>x. (f x, g x))"
unfolding eq
proof (rule proper_map_compose)
show "proper_map X (prod_topology Y X) (\<lambda>x. (f x,x))"
by (simp add: \<section> proper_map_paired_continuous_map_left)
show "proper_map (prod_topology Y X) (prod_topology Y Z) (\<lambda>(x,y). (x,g y))"
by (simp add: \<section> proper_map_prod proper_map_id [unfolded id_def])
qed
next
assume \<section>: "Hausdorff_space Z" "proper_map X Y f" "continuous_map X Z g"
have eq: "(\<lambda>x. (f x, g x)) = (\<lambda>(x,y). (f x,y)) \<circ> (\<lambda>x. (x,g x))"
by auto
show "proper_map X (prod_topology Y Z) (\<lambda>x. (f x, g x))"
unfolding eq
proof (rule proper_map_compose)
show "proper_map X (prod_topology X Z) (\<lambda>x. (x, g x))"
using \<section> proper_map_paired_continuous_map_right by auto
show "proper_map (prod_topology X Z) (prod_topology Y Z) (\<lambda>(x,y). (f x,y))"
by (simp add: \<section> proper_map_prod proper_map_id [unfolded id_def])
qed
qed
lemma proper_map_pairwise:
assumes
"Hausdorff_space X \<and> proper_map X Y (fst \<circ> f) \<and> proper_map X Z (snd \<circ> f) \<or>
Hausdorff_space Y \<and> continuous_map X Y (fst \<circ> f) \<and> proper_map X Z (snd \<circ> f) \<or>
Hausdorff_space Z \<and> proper_map X Y (fst \<circ> f) \<and> continuous_map X Z (snd \<circ> f)"
shows "proper_map X (prod_topology Y Z) f"
using proper_map_paired [OF assms] by (simp add: o_def)
lemma proper_map_from_composition_right:
assumes "Hausdorff_space Y" "proper_map X Z (g \<circ> f)" and contf: "continuous_map X Y f"
and contg: "continuous_map Y Z g"
shows "proper_map X Y f"
proof -
define YZ where "YZ \<equiv> subtopology (prod_topology Y Z) ((\<lambda>x. (x, g x)) ` topspace Y)"
have "proper_map X Y (fst \<circ> (\<lambda>x. (f x, (g \<circ> f) x)))"
proof (rule proper_map_compose)
have [simp]: "x \<in> topspace X \<Longrightarrow> f x \<in> topspace Y" for x
using contf continuous_map_preimage_topspace by auto
show "proper_map X YZ (\<lambda>x. (f x, (g \<circ> f) x))"
unfolding YZ_def
using assms
by (force intro!: proper_map_into_subtopology proper_map_paired simp: o_def image_iff)+
show "proper_map YZ Y fst"
using contg
by (simp flip: homeomorphic_maps_graph add: YZ_def homeomorphic_maps_map homeomorphic_imp_proper_map)
qed
moreover have "fst \<circ> (\<lambda>x. (f x, (g \<circ> f) x)) = f"
by auto
ultimately show ?thesis
by auto
qed
lemma perfect_map_from_composition_right:
"\<lbrakk>Hausdorff_space Y; perfect_map X Z (g \<circ> f);
continuous_map X Y f; continuous_map Y Z g; f ` topspace X = topspace Y\<rbrakk>
\<Longrightarrow> perfect_map X Y f"
by (meson perfect_map_def proper_map_from_composition_right)
lemma perfect_map_from_composition_right_inj:
"\<lbrakk>perfect_map X Z (g \<circ> f); f ` topspace X = topspace Y;
continuous_map X Y f; continuous_map Y Z g; inj_on g (topspace Y)\<rbrakk>
\<Longrightarrow> perfect_map X Y f"
by (meson continuous_map_openin_preimage_eq perfect_map_def proper_map_from_composition_right_inj)
subsection \<open>Regular spaces\<close>
text \<open>Regular spaces are *not* a priori assumed to be Hausdorff or $T_1$\<close>
definition regular_space
where "regular_space X \<equiv>
\<forall>C a. closedin X C \<and> a \<in> topspace X - C
\<longrightarrow> (\<exists>U V. openin X U \<and> openin X V \<and> a \<in> U \<and> C \<subseteq> V \<and> disjnt U V)"
lemma homeomorphic_regular_space_aux:
assumes hom: "X homeomorphic_space Y" and X: "regular_space X"
shows "regular_space Y"
proof -
obtain f g where hmf: "homeomorphic_map X Y f" and hmg: "homeomorphic_map Y X g"
and fg: "(\<forall>x \<in> topspace X. g(f x) = x) \<and> (\<forall>y \<in> topspace Y. f(g y) = y)"
using assms X homeomorphic_maps_map homeomorphic_space_def by fastforce
show ?thesis
unfolding regular_space_def
proof clarify
fix C a
assume Y: "closedin Y C" "a \<in> topspace Y" and "a \<notin> C"
then obtain "closedin X (g ` C)" "g a \<in> topspace X" "g a \<notin> g ` C"
using \<open>closedin Y C\<close> hmg homeomorphic_map_closedness_eq
by (smt (verit, ccfv_SIG) fg homeomorphic_imp_surjective_map image_iff in_mono)
then obtain S T where ST: "openin X S" "openin X T" "g a \<in> S" "g`C \<subseteq> T" "disjnt S T"
using X unfolding regular_space_def by (metis DiffI)
then have "openin Y (f`S)" "openin Y (f`T)"
by (meson hmf homeomorphic_map_openness_eq)+
moreover have "a \<in> f`S \<and> C \<subseteq> f`T"
using ST by (smt (verit, best) Y closedin_subset fg image_eqI subset_iff)
moreover have "disjnt (f`S) (f`T)"
using ST by (smt (verit, ccfv_SIG) disjnt_iff fg image_iff openin_subset subsetD)
ultimately show "\<exists>U V. openin Y U \<and> openin Y V \<and> a \<in> U \<and> C \<subseteq> V \<and> disjnt U V"
by metis
qed
qed
lemma homeomorphic_regular_space:
"X homeomorphic_space Y
\<Longrightarrow> (regular_space X \<longleftrightarrow> regular_space Y)"
by (meson homeomorphic_regular_space_aux homeomorphic_space_sym)
lemma regular_space:
"regular_space X \<longleftrightarrow>
(\<forall>C a. closedin X C \<and> a \<in> topspace X - C
\<longrightarrow> (\<exists>U. openin X U \<and> a \<in> U \<and> disjnt C (X closure_of U)))"
unfolding regular_space_def
proof (intro all_cong1 imp_cong refl ex_cong1)
fix C a U
assume C: "closedin X C \<and> a \<in> topspace X - C"
show "(\<exists>V. openin X U \<and> openin X V \<and> a \<in> U \<and> C \<subseteq> V \<and> disjnt U V)
\<longleftrightarrow> (openin X U \<and> a \<in> U \<and> disjnt C (X closure_of U))" (is "?lhs=?rhs")
proof
assume ?lhs
then show ?rhs
by (smt (verit, best) disjnt_iff in_closure_of subsetD)
next
assume R: ?rhs
then have "disjnt U (topspace X - X closure_of U)"
by (meson DiffD2 closure_of_subset disjnt_iff openin_subset subsetD)
moreover have "C \<subseteq> topspace X - X closure_of U"
by (meson C DiffI R closedin_subset disjnt_iff subset_eq)
ultimately show ?lhs
using R by (rule_tac x="topspace X - X closure_of U" in exI) auto
qed
qed
lemma neighbourhood_base_of_closedin:
"neighbourhood_base_of (closedin X) X \<longleftrightarrow> regular_space X" (is "?lhs=?rhs")
proof -
have "?lhs \<longleftrightarrow> (\<forall>W x. openin X W \<and> x \<in> W \<longrightarrow>
(\<exists>U. openin X U \<and> (\<exists>V. closedin X V \<and> x \<in> U \<and> U \<subseteq> V \<and> V \<subseteq> W)))"
by (simp add: neighbourhood_base_of)
also have "\<dots> \<longleftrightarrow> (\<forall>W x. closedin X W \<and> x \<in> topspace X - W \<longrightarrow>
(\<exists>U. openin X U \<and> (\<exists>V. closedin X V \<and> x \<in> U \<and> U \<subseteq> V \<and> V \<subseteq> topspace X - W)))"
by (smt (verit) Diff_Diff_Int closedin_def inf.absorb_iff2 openin_closedin_eq)
also have "\<dots> \<longleftrightarrow> ?rhs"
proof -
have \<section>: "(\<exists>V. closedin X V \<and> x \<in> U \<and> U \<subseteq> V \<and> V \<subseteq> topspace X - W)
\<longleftrightarrow> (\<exists>V. openin X V \<and> x \<in> U \<and> W \<subseteq> V \<and> disjnt U V)" (is "?lhs=?rhs")
if "openin X U" "closedin X W" "x \<in> topspace X" "x \<notin> W" for W U x
proof
assume ?lhs with \<open>closedin X W\<close> show ?rhs
unfolding closedin_def by (smt (verit) Diff_mono disjnt_Diff1 double_diff subset_eq)
next
assume ?rhs with \<open>openin X U\<close> show ?lhs
unfolding openin_closedin_eq disjnt_def
by (smt (verit) Diff_Diff_Int Diff_disjoint Diff_eq_empty_iff Int_Diff inf.orderE)
qed
show ?thesis
unfolding regular_space_def
by (intro all_cong1 ex_cong1 imp_cong refl) (metis \<section> DiffE)
qed
finally show ?thesis .
qed
lemma regular_space_discrete_topology [simp]:
"regular_space (discrete_topology S)"
using neighbourhood_base_of_closedin neighbourhood_base_of_discrete_topology by fastforce
lemma regular_space_subtopology:
"regular_space X \<Longrightarrow> regular_space (subtopology X S)"
unfolding regular_space_def openin_subtopology_alt closedin_subtopology_alt disjnt_iff
by clarsimp (smt (verit, best) inf.orderE inf_le1 le_inf_iff)
lemma regular_space_retraction_map_image:
"\<lbrakk>retraction_map X Y r; regular_space X\<rbrakk> \<Longrightarrow> regular_space Y"
using hereditary_imp_retractive_property homeomorphic_regular_space regular_space_subtopology by blast
lemma regular_t0_imp_Hausdorff_space:
"\<lbrakk>regular_space X; t0_space X\<rbrakk> \<Longrightarrow> Hausdorff_space X"
apply (clarsimp simp: regular_space_def t0_space Hausdorff_space_def)
by (metis disjnt_sym subsetD)
lemma regular_t0_eq_Hausdorff_space:
"regular_space X \<Longrightarrow> (t0_space X \<longleftrightarrow> Hausdorff_space X)"
using Hausdorff_imp_t0_space regular_t0_imp_Hausdorff_space by blast
lemma regular_t1_imp_Hausdorff_space:
"\<lbrakk>regular_space X; t1_space X\<rbrakk> \<Longrightarrow> Hausdorff_space X"
by (simp add: regular_t0_imp_Hausdorff_space t1_imp_t0_space)
lemma regular_t1_eq_Hausdorff_space:
"regular_space X \<Longrightarrow> t1_space X \<longleftrightarrow> Hausdorff_space X"
using regular_t0_imp_Hausdorff_space t1_imp_t0_space t1_or_Hausdorff_space by blast
lemma compact_Hausdorff_imp_regular_space:
assumes "compact_space X" "Hausdorff_space X"
shows "regular_space X"
unfolding regular_space_def
proof clarify
fix S a
assume "closedin X S" and "a \<in> topspace X" and "a \<notin> S"
then show "\<exists>U V. openin X U \<and> openin X V \<and> a \<in> U \<and> S \<subseteq> V \<and> disjnt U V"
using assms unfolding Hausdorff_space_compact_sets
by (metis closedin_compact_space compactin_sing disjnt_empty1 insert_subset disjnt_insert1)
qed
lemma neighbourhood_base_of_closed_Hausdorff_space:
"regular_space X \<and> Hausdorff_space X \<longleftrightarrow>
neighbourhood_base_of (\<lambda>C. closedin X C \<and> Hausdorff_space(subtopology X C)) X" (is "?lhs=?rhs")
proof
assume ?lhs then show ?rhs
by (simp add: Hausdorff_space_subtopology neighbourhood_base_of_closedin)
next
assume ?rhs then show ?lhs
by (metis (mono_tags, lifting) Hausdorff_space_closed_neighbourhood neighbourhood_base_of neighbourhood_base_of_closedin openin_topspace)
qed
lemma locally_compact_imp_kc_eq_Hausdorff_space:
"neighbourhood_base_of (compactin X) X \<Longrightarrow> kc_space X \<longleftrightarrow> Hausdorff_space X"
by (metis Hausdorff_imp_kc_space kc_imp_t1_space kc_space_def neighbourhood_base_of_closedin neighbourhood_base_of_mono regular_t1_imp_Hausdorff_space)
lemma regular_space_compact_closed_separation:
assumes X: "regular_space X"
and S: "compactin X S"
and T: "closedin X T"
and "disjnt S T"
shows "\<exists>U V. openin X U \<and> openin X V \<and> S \<subseteq> U \<and> T \<subseteq> V \<and> disjnt U V"
proof (cases "S={}")
case True
then show ?thesis
by (meson T closedin_def disjnt_empty1 empty_subsetI openin_empty openin_topspace)
next
case False
then have "\<exists>U V. x \<in> S \<longrightarrow> openin X U \<and> openin X V \<and> x \<in> U \<and> T \<subseteq> V \<and> disjnt U V" for x
using assms unfolding regular_space_def
by (smt (verit) Diff_iff compactin_subset_topspace disjnt_iff subsetD)
then obtain U V where UV: "\<And>x. x \<in> S \<Longrightarrow> openin X (U x) \<and> openin X (V x) \<and> x \<in> (U x) \<and> T \<subseteq> (V x) \<and> disjnt (U x) (V x)"
by metis
then obtain \<F> where "finite \<F>" "\<F> \<subseteq> U ` S" "S \<subseteq> \<Union> \<F>"
using S unfolding compactin_def by (smt (verit) UN_iff image_iff subsetI)
then obtain K where "finite K" "K \<subseteq> S" and K: "S \<subseteq> \<Union>(U ` K)"
by (metis finite_subset_image)
show ?thesis
proof (intro exI conjI)
show "openin X (\<Union>(U ` K))"
using \<open>K \<subseteq> S\<close> UV by blast
show "openin X (\<Inter>(V ` K))"
using False K UV \<open>K \<subseteq> S\<close> \<open>finite K\<close> by blast
show "S \<subseteq> \<Union>(U ` K)"
by (simp add: K)
show "T \<subseteq> \<Inter>(V ` K)"
using UV \<open>K \<subseteq> S\<close> by blast
show "disjnt (\<Union>(U ` K)) (\<Inter>(V ` K))"
by (smt (verit) Inter_iff UN_E UV \<open>K \<subseteq> S\<close> disjnt_iff image_eqI subset_iff)
qed
qed
lemma regular_space_compact_closed_sets:
"regular_space X \<longleftrightarrow>
(\<forall>S T. compactin X S \<and> closedin X T \<and> disjnt S T
\<longrightarrow> (\<exists>U V. openin X U \<and> openin X V \<and> S \<subseteq> U \<and> T \<subseteq> V \<and> disjnt U V))" (is "?lhs=?rhs")
proof
assume ?lhs then show ?rhs
using regular_space_compact_closed_separation by fastforce
next
assume R: ?rhs
show ?lhs
unfolding regular_space
proof clarify
fix S x
assume "closedin X S" and "x \<in> topspace X" and "x \<notin> S"
then obtain U V where "openin X U \<and> openin X V \<and> {x} \<subseteq> U \<and> S \<subseteq> V \<and> disjnt U V"
by (metis R compactin_sing disjnt_empty1 disjnt_insert1)
then show "\<exists>U. openin X U \<and> x \<in> U \<and> disjnt S (X closure_of U)"
by (smt (verit, best) disjnt_iff in_closure_of insert_subset subsetD)
qed
qed
lemma regular_space_prod_topology:
"regular_space (prod_topology X Y) \<longleftrightarrow>
X = trivial_topology \<or> Y = trivial_topology \<or> regular_space X \<and> regular_space Y" (is "?lhs=?rhs")
proof
assume ?lhs
then show ?rhs
by (metis regular_space_retraction_map_image retraction_map_fst retraction_map_snd)
next
assume R: ?rhs
show ?lhs
proof (cases "X = trivial_topology \<or> Y = trivial_topology")
case True then show ?thesis by auto
next
case False
then have "regular_space X" "regular_space Y"
using R by auto
show ?thesis
unfolding neighbourhood_base_of_closedin [symmetric] neighbourhood_base_of
proof clarify
fix W x y
assume W: "openin (prod_topology X Y) W" and "(x,y) \<in> W"
then obtain U V where U: "openin X U" "x \<in> U" and V: "openin Y V" "y \<in> V"
and "U \<times> V \<subseteq> W"
by (metis openin_prod_topology_alt)
obtain D1 C1 where 1: "openin X D1" "closedin X C1" "x \<in> D1" "D1 \<subseteq> C1" "C1 \<subseteq> U"
by (metis \<open>regular_space X\<close> U neighbourhood_base_of neighbourhood_base_of_closedin)
obtain D2 C2 where 2: "openin Y D2" "closedin Y C2" "y \<in> D2" "D2 \<subseteq> C2" "C2 \<subseteq> V"
by (metis \<open>regular_space Y\<close> V neighbourhood_base_of neighbourhood_base_of_closedin)
show "\<exists>U V. openin (prod_topology X Y) U \<and> closedin (prod_topology X Y) V \<and>
(x,y) \<in> U \<and> U \<subseteq> V \<and> V \<subseteq> W"
proof (intro conjI exI)
show "openin (prod_topology X Y) (D1 \<times> D2)"
by (simp add: 1 2 openin_prod_Times_iff)
show "closedin (prod_topology X Y) (C1 \<times> C2)"
by (simp add: 1 2 closedin_prod_Times_iff)
qed (use 1 2 \<open>U \<times> V \<subseteq> W\<close> in auto)
qed
qed
qed
lemma regular_space_product_topology:
"regular_space (product_topology X I) \<longleftrightarrow>
(product_topology X I) = trivial_topology \<or> (\<forall>i \<in> I. regular_space (X i))" (is "?lhs=?rhs")
proof
assume ?lhs
then show ?rhs
by (meson regular_space_retraction_map_image retraction_map_product_projection)
next
assume R: ?rhs
show ?lhs
proof (cases "product_topology X I = trivial_topology")
case True
then show ?thesis
by auto
next
case False
then obtain x where x: "x \<in> topspace (product_topology X I)"
by (meson ex_in_conv null_topspace_iff_trivial)
define \<F> where "\<F> \<equiv> {Pi\<^sub>E I U |U. finite {i \<in> I. U i \<noteq> topspace (X i)}
\<and> (\<forall>i\<in>I. openin (X i) (U i))}"
have oo: "openin (product_topology X I) = arbitrary union_of (\<lambda>W. W \<in> \<F>)"
by (simp add: \<F>_def openin_product_topology product_topology_base_alt)
have "\<exists>U V. openin (product_topology X I) U \<and> closedin (product_topology X I) V \<and> x \<in> U \<and> U \<subseteq> V \<and> V \<subseteq> Pi\<^sub>E I W"
if fin: "finite {i \<in> I. W i \<noteq> topspace (X i)}"
and opeW: "\<And>k. k \<in> I \<Longrightarrow> openin (X k) (W k)" and x: "x \<in> PiE I W" for W x
proof -
have "\<And>i. i \<in> I \<Longrightarrow> \<exists>U V. openin (X i) U \<and> closedin (X i) V \<and> x i \<in> U \<and> U \<subseteq> V \<and> V \<subseteq> W i"
by (metis False PiE_iff R neighbourhood_base_of neighbourhood_base_of_closedin opeW x)
then obtain U C where UC:
"\<And>i. i \<in> I \<Longrightarrow> openin (X i) (U i) \<and> closedin (X i) (C i) \<and> x i \<in> U i \<and> U i \<subseteq> C i \<and> C i \<subseteq> W i"
by metis
define PI where "PI \<equiv> \<lambda>V. PiE I (\<lambda>i. if W i = topspace(X i) then topspace(X i) else V i)"
show ?thesis
proof (intro conjI exI)
have "\<forall>i\<in>I. W i \<noteq> topspace (X i) \<longrightarrow> openin (X i) (U i)"
using UC by force
with fin show "openin (product_topology X I) (PI U)"
by (simp add: Collect_mono_iff PI_def openin_PiE_gen rev_finite_subset)
show "closedin (product_topology X I) (PI C)"
by (simp add: UC closedin_product_topology PI_def)
show "x \<in> PI U"
using UC x by (fastforce simp: PI_def)
show "PI U \<subseteq> PI C"
by (smt (verit) UC Orderings.order_eq_iff PiE_mono PI_def)
show "PI C \<subseteq> Pi\<^sub>E I W"
by (simp add: UC PI_def subset_PiE)
qed
qed
then have "neighbourhood_base_of (closedin (product_topology X I)) (product_topology X I)"
unfolding neighbourhood_base_of_topology_base [OF oo] by (force simp: \<F>_def)
then show ?thesis
by (simp add: neighbourhood_base_of_closedin)
qed
qed
lemma closed_map_paired_gen_aux:
assumes "regular_space Y" and f: "closed_map Z X f" and g: "closed_map Z Y g"
and clo: "\<And>y. y \<in> topspace X \<Longrightarrow> closedin Z {x \<in> topspace Z. f x = y}"
and contg: "continuous_map Z Y g"
shows "closed_map Z (prod_topology X Y) (\<lambda>x. (f x, g x))"
unfolding closed_map_def
proof (intro strip)
fix C assume "closedin Z C"
then have "C \<subseteq> topspace Z"
by (simp add: closedin_subset)
have "f \<in> topspace Z \<rightarrow> topspace X" "g \<in> topspace Z \<rightarrow> topspace Y"
by (simp_all add: assms closed_map_imp_subset_topspace)
show "closedin (prod_topology X Y) ((\<lambda>x. (f x, g x)) ` C)"
unfolding closedin_def topspace_prod_topology
proof (intro conjI)
have "closedin Y (g ` C)"
using \<open>closedin Z C\<close> assms(3) closed_map_def by blast
with assms show "(\<lambda>x. (f x, g x)) ` C \<subseteq> topspace X \<times> topspace Y"
by (smt (verit) SigmaI \<open>closedin Z C\<close> closed_map_def closedin_subset image_subset_iff)
have *: "\<exists>T. openin (prod_topology X Y) T \<and> (y1,y2) \<in> T \<and> T \<subseteq> topspace X \<times> topspace Y - (\<lambda>x. (f x, g x)) ` C"
if "(y1,y2) \<notin> (\<lambda>x. (f x, g x)) ` C" and y1: "y1 \<in> topspace X" and y2: "y2 \<in> topspace Y"
for y1 y2
proof -
define A where "A \<equiv> topspace Z - (C \<inter> {x \<in> topspace Z. f x = y1})"
have A: "openin Z A" "{x \<in> topspace Z. g x = y2} \<subseteq> A"
using that \<open>closedin Z C\<close> clo that(2) by (auto simp: A_def)
obtain V0 where "openin Y V0 \<and> y2 \<in> V0" and UA: "{x \<in> topspace Z. g x \<in> V0} \<subseteq> A"
using g A y2 unfolding closed_map_fibre_neighbourhood by blast
then obtain V V' where VV: "openin Y V \<and> closedin Y V' \<and> y2 \<in> V \<and> V \<subseteq> V'" and "V' \<subseteq> V0"
by (metis (no_types, lifting) \<open>regular_space Y\<close> neighbourhood_base_of neighbourhood_base_of_closedin)
with UA have subA: "{x \<in> topspace Z. g x \<in> V'} \<subseteq> A"
by blast
show ?thesis
proof -
define B where "B \<equiv> topspace Z - (C \<inter> {x \<in> topspace Z. g x \<in> V'})"
have "openin Z B"
using VV \<open>closedin Z C\<close> contg by (fastforce simp: B_def continuous_map_closedin)
have "{x \<in> topspace Z. f x = y1} \<subseteq> B"
using A_def subA by (auto simp: A_def B_def)
then obtain U where "openin X U" "y1 \<in> U" and subB: "{x \<in> topspace Z. f x \<in> U} \<subseteq> B"
using \<open>openin Z B\<close> y1 f unfolding closed_map_fibre_neighbourhood by meson
show ?thesis
proof (intro conjI exI)
show "openin (prod_topology X Y) (U \<times> V)"
by (metis VV \<open>openin X U\<close> openin_prod_Times_iff)
show "(y1, y2) \<in> U \<times> V"
by (simp add: VV \<open>y1 \<in> U\<close>)
show "U \<times> V \<subseteq> topspace X \<times> topspace Y - (\<lambda>x. (f x, g x)) ` C"
using VV \<open>C \<subseteq> topspace Z\<close> \<open>openin X U\<close> subB
by (force simp: image_iff B_def subset_iff dest: openin_subset)
qed
qed
qed
then show "openin (prod_topology X Y) (topspace X \<times> topspace Y - (\<lambda>x. (f x, g x)) ` C)"
by (smt (verit, ccfv_threshold) Diff_iff SigmaE openin_subopen)
qed
qed
lemma closed_map_paired_gen:
assumes f: "closed_map Z X f" and g: "closed_map Z Y g"
and D: "(regular_space X \<and> continuous_map Z X f \<and> (\<forall>z \<in> topspace Y. closedin Z {x \<in> topspace Z. g x = z})
\<or> regular_space Y \<and> continuous_map Z Y g \<and> (\<forall>y \<in> topspace X. closedin Z {x \<in> topspace Z. f x = y}))"
(is "?RSX \<or> ?RSY")
shows "closed_map Z (prod_topology X Y) (\<lambda>x. (f x, g x))"
using D
proof
assume RSX: ?RSX
have eq: "(\<lambda>x. (f x, g x)) = (\<lambda>(x,y). (y,x)) \<circ> (\<lambda>x. (g x, f x))"
by auto
show ?thesis
unfolding eq
proof (rule closed_map_compose)
show "closed_map Z (prod_topology Y X) (\<lambda>x. (g x, f x))"
using RSX closed_map_paired_gen_aux f g by fastforce
show "closed_map (prod_topology Y X) (prod_topology X Y) (\<lambda>(x, y). (y, x))"
using homeomorphic_imp_closed_map homeomorphic_map_swap by blast
qed
qed (blast intro: assms closed_map_paired_gen_aux)
lemma closed_map_paired:
assumes "closed_map Z X f" and contf: "continuous_map Z X f"
"closed_map Z Y g" and contg: "continuous_map Z Y g"
and D: "t1_space X \<and> regular_space Y \<or> regular_space X \<and> t1_space Y"
shows "closed_map Z (prod_topology X Y) (\<lambda>x. (f x, g x))"
proof (rule closed_map_paired_gen)
show "regular_space X \<and> continuous_map Z X f \<and> (\<forall>z\<in>topspace Y. closedin Z {x \<in> topspace Z. g x = z}) \<or> regular_space Y \<and> continuous_map Z Y g \<and> (\<forall>y\<in>topspace X. closedin Z {x \<in> topspace Z. f x = y})"
using D contf contg
by (smt (verit, del_insts) Collect_cong closedin_continuous_map_preimage t1_space_closedin_singleton singleton_iff)
qed (use assms in auto)
lemma closed_map_pairwise:
assumes "closed_map Z X (fst \<circ> f)" "continuous_map Z X (fst \<circ> f)"
"closed_map Z Y (snd \<circ> f)" "continuous_map Z Y (snd \<circ> f)"
"t1_space X \<and> regular_space Y \<or> regular_space X \<and> t1_space Y"
shows "closed_map Z (prod_topology X Y) f"
proof -
have "closed_map Z (prod_topology X Y) (\<lambda>a. ((fst \<circ> f) a, (snd \<circ> f) a))"
using assms closed_map_paired by blast
then show ?thesis
by auto
qed
lemma continuous_imp_proper_map:
"\<lbrakk>compact_space X; kc_space Y; continuous_map X Y f\<rbrakk> \<Longrightarrow> proper_map X Y f"
by (simp add: continuous_closed_imp_proper_map continuous_imp_closed_map_gen kc_imp_t1_space)
lemma tube_lemma_right:
assumes W: "openin (prod_topology X Y) W" and C: "compactin Y C"
and x: "x \<in> topspace X" and subW: "{x} \<times> C \<subseteq> W"
shows "\<exists>U V. openin X U \<and> openin Y V \<and> x \<in> U \<and> C \<subseteq> V \<and> U \<times> V \<subseteq> W"
proof (cases "C = {}")
case True
with x show ?thesis by auto
next
case False
have "\<exists>U V. openin X U \<and> openin Y V \<and> x \<in> U \<and> y \<in> V \<and> U \<times> V \<subseteq> W"
if "y \<in> C" for y
using W openin_prod_topology_alt subW subsetD that by fastforce
then obtain U V where UV: "\<And>y. y \<in> C \<Longrightarrow> openin X (U y) \<and> openin Y (V y) \<and> x \<in> U y \<and> y \<in> V y \<and> U y \<times> V y \<subseteq> W"
by metis
then obtain D where D: "finite D" "D \<subseteq> C" "C \<subseteq> \<Union> (V ` D)"
using compactinD [OF C, of "V`C"]
by (smt (verit) UN_I finite_subset_image imageE subsetI)
show ?thesis
proof (intro exI conjI)
show "openin X (\<Inter> (U ` D))" "openin Y (\<Union> (V ` D))"
using D False UV by blast+
show "x \<in> \<Inter> (U ` D)" "C \<subseteq> \<Union> (V ` D)" "\<Inter> (U ` D) \<times> \<Union> (V ` D) \<subseteq> W"
using D UV by force+
qed
qed
lemma closed_map_fst:
assumes "compact_space Y"
shows "closed_map (prod_topology X Y) X fst"
proof -
have *: "{x \<in> topspace X \<times> topspace Y. fst x \<in> U} = U \<times> topspace Y"
if "U \<subseteq> topspace X" for U
using that by force
have **: "\<And>U y. \<lbrakk>openin (prod_topology X Y) U; y \<in> topspace X;
{x \<in> topspace X \<times> topspace Y. fst x = y} \<subseteq> U\<rbrakk>
\<Longrightarrow> \<exists>V. openin X V \<and> y \<in> V \<and> V \<times> topspace Y \<subseteq> U"
using tube_lemma_right[of X Y _ "topspace Y"] assms by (fastforce simp: compact_space_def)
show ?thesis
unfolding closed_map_fibre_neighbourhood
by (force simp: * openin_subset cong: conj_cong intro: **)
qed
lemma closed_map_snd:
assumes "compact_space X"
shows "closed_map (prod_topology X Y) Y snd"
proof -
have "snd = fst o prod.swap"
by force
moreover have "closed_map (prod_topology X Y) Y (fst o prod.swap)"
proof (rule closed_map_compose)
show "closed_map (prod_topology X Y) (prod_topology Y X) prod.swap"
by (metis (no_types, lifting) homeomorphic_imp_closed_map homeomorphic_map_eq homeomorphic_map_swap prod.swap_def split_beta)
show "closed_map (prod_topology Y X) Y fst"
by (simp add: closed_map_fst assms)
qed
ultimately show ?thesis
by metis
qed
lemma closed_map_paired_closed_map_right:
"\<lbrakk>closed_map X Y f; regular_space X;
\<And>y. y \<in> topspace Y \<Longrightarrow> closedin X {x \<in> topspace X. f x = y}\<rbrakk>
\<Longrightarrow> closed_map X (prod_topology X Y) (\<lambda>x. (x, f x))"
by (rule closed_map_paired_gen [OF closed_map_id, unfolded id_def]) auto
lemma closed_map_paired_closed_map_left:
assumes "closed_map X Y f" "regular_space X"
"\<And>y. y \<in> topspace Y \<Longrightarrow> closedin X {x \<in> topspace X. f x = y}"
shows "closed_map X (prod_topology Y X) (\<lambda>x. (f x, x))"
proof -
have eq: "(\<lambda>x. (f x, x)) = (\<lambda>(x,y). (y,x)) \<circ> (\<lambda>x. (x, f x))"
by auto
show ?thesis
unfolding eq
proof (rule closed_map_compose)
show "closed_map X (prod_topology X Y) (\<lambda>x. (x, f x))"
by (simp add: assms closed_map_paired_closed_map_right)
show "closed_map (prod_topology X Y) (prod_topology Y X) (\<lambda>(x, y). (y, x))"
using homeomorphic_imp_closed_map homeomorphic_map_swap by blast
qed
qed
lemma closed_map_imp_closed_graph:
assumes "closed_map X Y f" "regular_space X"
"\<And>y. y \<in> topspace Y \<Longrightarrow> closedin X {x \<in> topspace X. f x = y}"
shows "closedin (prod_topology X Y) ((\<lambda>x. (x, f x)) ` topspace X)"
using assms closed_map_def closed_map_paired_closed_map_right by blast
lemma proper_map_paired_closed_map_right:
assumes "closed_map X Y f" "regular_space X"
"\<And>y. y \<in> topspace Y \<Longrightarrow> closedin X {x \<in> topspace X. f x = y}"
shows "proper_map X (prod_topology X Y) (\<lambda>x. (x, f x))"
by (simp add: assms closed_injective_imp_proper_map inj_on_def closed_map_paired_closed_map_right)
lemma proper_map_paired_closed_map_left:
assumes "closed_map X Y f" "regular_space X"
"\<And>y. y \<in> topspace Y \<Longrightarrow> closedin X {x \<in> topspace X. f x = y}"
shows "proper_map X (prod_topology Y X) (\<lambda>x. (f x, x))"
by (simp add: assms closed_injective_imp_proper_map inj_on_def closed_map_paired_closed_map_left)
proposition regular_space_continuous_proper_map_image:
assumes "regular_space X" and contf: "continuous_map X Y f" and pmapf: "proper_map X Y f"
and fim: "f ` (topspace X) = topspace Y"
shows "regular_space Y"
unfolding regular_space_def
proof clarify
fix C y
assume "closedin Y C" and "y \<in> topspace Y" and "y \<notin> C"
have "closed_map X Y f" "(\<forall>y \<in> topspace Y. compactin X {x \<in> topspace X. f x = y})"
using pmapf proper_map_def by force+
moreover have "closedin X {z \<in> topspace X. f z \<in> C}"
using \<open>closedin Y C\<close> contf continuous_map_closedin by fastforce
moreover have "disjnt {z \<in> topspace X. f z = y} {z \<in> topspace X. f z \<in> C}"
using \<open>y \<notin> C\<close> disjnt_iff by blast
ultimately
obtain U V where UV: "openin X U" "openin X V" "{z \<in> topspace X. f z = y} \<subseteq> U" "{z \<in> topspace X. f z \<in> C} \<subseteq> V"
and dUV: "disjnt U V"
using \<open>y \<in> topspace Y\<close> \<open>regular_space X\<close> unfolding regular_space_compact_closed_sets
by meson
have *: "\<And>U T. openin X U \<and> T \<subseteq> topspace Y \<and> {x \<in> topspace X. f x \<in> T} \<subseteq> U \<longrightarrow>
(\<exists>V. openin Y V \<and> T \<subseteq> V \<and> {x \<in> topspace X. f x \<in> V} \<subseteq> U)"
using \<open>closed_map X Y f\<close> unfolding closed_map_preimage_neighbourhood by blast
obtain V1 where V1: "openin Y V1 \<and> y \<in> V1" and sub1: "{x \<in> topspace X. f x \<in> V1} \<subseteq> U"
using * [of U "{y}"] UV \<open>y \<in> topspace Y\<close> by auto
moreover
obtain V2 where "openin Y V2 \<and> C \<subseteq> V2" and sub2: "{x \<in> topspace X. f x \<in> V2} \<subseteq> V"
by (smt (verit, ccfv_SIG) * UV \<open>closedin Y C\<close> closedin_subset mem_Collect_eq subset_iff)
moreover have "disjnt V1 V2"
proof -
have "\<And>x. \<lbrakk>\<forall>x. x \<in> U \<longrightarrow> x \<notin> V; x \<in> V1; x \<in> V2\<rbrakk> \<Longrightarrow> False"
by (smt (verit) V1 fim image_iff mem_Collect_eq openin_subset sub1 sub2 subsetD)
with dUV show ?thesis by (auto simp: disjnt_iff)
qed
ultimately show "\<exists>U V. openin Y U \<and> openin Y V \<and> y \<in> U \<and> C \<subseteq> V \<and> disjnt U V"
by meson
qed
lemma regular_space_perfect_map_image:
"\<lbrakk>regular_space X; perfect_map X Y f\<rbrakk> \<Longrightarrow> regular_space Y"
by (meson perfect_map_def regular_space_continuous_proper_map_image)
proposition regular_space_perfect_map_image_eq:
assumes "Hausdorff_space X" and perf: "perfect_map X Y f"
shows "regular_space X \<longleftrightarrow> regular_space Y" (is "?lhs=?rhs")
proof
assume ?lhs
then show ?rhs
using perf regular_space_perfect_map_image by blast
next
assume R: ?rhs
have "continuous_map X Y f" "proper_map X Y f" and fim: "f ` (topspace X) = topspace Y"
using perf by (auto simp: perfect_map_def)
then have "closed_map X Y f" and preYf: "(\<forall>y \<in> topspace Y. compactin X {x \<in> topspace X. f x = y})"
by (simp_all add: proper_map_def)
show ?lhs
unfolding regular_space_def
proof clarify
fix C x
assume "closedin X C" and "x \<in> topspace X" and "x \<notin> C"
obtain U1 U2 where "openin X U1" "openin X U2" "{x} \<subseteq> U1" and "disjnt U1 U2"
and subV: "C \<inter> {z \<in> topspace X. f z = f x} \<subseteq> U2"
proof (rule Hausdorff_space_compact_separation [of X "{x}" "C \<inter> {z \<in> topspace X. f z = f x}", OF \<open>Hausdorff_space X\<close>])
show "compactin X {x}"
by (simp add: \<open>x \<in> topspace X\<close>)
show "compactin X (C \<inter> {z \<in> topspace X. f z = f x})"
using \<open>closedin X C\<close> fim \<open>x \<in> topspace X\<close> closed_Int_compactin preYf by fastforce
show "disjnt {x} (C \<inter> {z \<in> topspace X. f z = f x})"
using \<open>x \<notin> C\<close> by force
qed
have "closedin Y (f ` (C - U2))"
using \<open>closed_map X Y f\<close> \<open>closedin X C\<close> \<open>openin X U2\<close> closed_map_def by blast
moreover
have "f x \<in> topspace Y - f ` (C - U2)"
using \<open>closedin X C\<close> \<open>continuous_map X Y f\<close> \<open>x \<in> topspace X\<close> closedin_subset continuous_map_def subV
by (fastforce simp: Pi_iff)
ultimately
obtain V1 V2 where VV: "openin Y V1" "openin Y V2" "f x \<in> V1"
and subV2: "f ` (C - U2) \<subseteq> V2" and "disjnt V1 V2"
by (meson R regular_space_def)
show "\<exists>U U'. openin X U \<and> openin X U' \<and> x \<in> U \<and> C \<subseteq> U' \<and> disjnt U U'"
proof (intro exI conjI)
show "openin X (U1 \<inter> {x \<in> topspace X. f x \<in> V1})"
using VV(1) \<open>continuous_map X Y f\<close> \<open>openin X U1\<close> continuous_map by fastforce
show "openin X (U2 \<union> {x \<in> topspace X. f x \<in> V2})"
using VV(2) \<open>continuous_map X Y f\<close> \<open>openin X U2\<close> continuous_map by fastforce
show "x \<in> U1 \<inter> {x \<in> topspace X. f x \<in> V1}"
using VV(3) \<open>x \<in> topspace X\<close> \<open>{x} \<subseteq> U1\<close> by auto
show "C \<subseteq> U2 \<union> {x \<in> topspace X. f x \<in> V2}"
using \<open>closedin X C\<close> closedin_subset subV2 by auto
show "disjnt (U1 \<inter> {x \<in> topspace X. f x \<in> V1}) (U2 \<union> {x \<in> topspace X. f x \<in> V2})"
using \<open>disjnt U1 U2\<close> \<open>disjnt V1 V2\<close> by (auto simp: disjnt_iff)
qed
qed
qed
subsection\<open>Locally compact spaces\<close>
definition locally_compact_space
where "locally_compact_space X \<equiv>
\<forall>x \<in> topspace X. \<exists>U K. openin X U \<and> compactin X K \<and> x \<in> U \<and> U \<subseteq> K"
lemma homeomorphic_locally_compact_spaceD:
assumes X: "locally_compact_space X" and "X homeomorphic_space Y"
shows "locally_compact_space Y"
proof -
obtain f where hmf: "homeomorphic_map X Y f"
using assms homeomorphic_space by blast
then have eq: "topspace Y = f ` (topspace X)"
by (simp add: homeomorphic_imp_surjective_map)
have "\<exists>V K. openin Y V \<and> compactin Y K \<and> f x \<in> V \<and> V \<subseteq> K"
if "x \<in> topspace X" "openin X U" "compactin X K" "x \<in> U" "U \<subseteq> K" for x U K
using that
by (meson hmf homeomorphic_map_compactness_eq homeomorphic_map_openness_eq image_mono image_eqI)
with X show ?thesis
by (smt (verit) eq image_iff locally_compact_space_def)
qed
lemma homeomorphic_locally_compact_space:
assumes "X homeomorphic_space Y"
shows "locally_compact_space X \<longleftrightarrow> locally_compact_space Y"
by (meson assms homeomorphic_locally_compact_spaceD homeomorphic_space_sym)
lemma locally_compact_space_retraction_map_image:
assumes "retraction_map X Y r" and X: "locally_compact_space X"
shows "locally_compact_space Y"
proof -
obtain s where s: "retraction_maps X Y r s"
using assms retraction_map_def by blast
obtain T where "T retract_of_space X" and Teq: "T = s ` topspace Y"
using retraction_maps_section_image1 s by blast
then obtain r where r: "continuous_map X (subtopology X T) r" "\<forall>x\<in>T. r x = x"
by (meson retract_of_space_def)
have "locally_compact_space (subtopology X T)"
unfolding locally_compact_space_def openin_subtopology_alt
proof clarsimp
fix x
assume "x \<in> topspace X" "x \<in> T"
obtain U K where UK: "openin X U \<and> compactin X K \<and> x \<in> U \<and> U \<subseteq> K"
by (meson X \<open>x \<in> topspace X\<close> locally_compact_space_def)
then have "compactin (subtopology X T) (r ` K) \<and> T \<inter> U \<subseteq> r ` K"
by (smt (verit) IntD1 image_compactin image_iff inf_le2 r subset_iff)
then show "\<exists>U. openin X U \<and> (\<exists>K. compactin (subtopology X T) K \<and> x \<in> U \<and> T \<inter> U \<subseteq> K)"
using UK by auto
qed
with Teq show ?thesis
using homeomorphic_locally_compact_space retraction_maps_section_image2 s by blast
qed
lemma compact_imp_locally_compact_space:
"compact_space X \<Longrightarrow> locally_compact_space X"
using compact_space_def locally_compact_space_def by blast
lemma neighbourhood_base_imp_locally_compact_space:
"neighbourhood_base_of (compactin X) X \<Longrightarrow> locally_compact_space X"
by (metis locally_compact_space_def neighbourhood_base_of openin_topspace)
lemma locally_compact_imp_neighbourhood_base:
assumes loc: "locally_compact_space X" and reg: "regular_space X"
shows "neighbourhood_base_of (compactin X) X"
unfolding neighbourhood_base_of
proof clarify
fix W x
assume "openin X W" and "x \<in> W"
then obtain U K where "openin X U" "compactin X K" "x \<in> U" "U \<subseteq> K"
by (metis loc locally_compact_space_def openin_subset subsetD)
moreover have "openin X (U \<inter> W) \<and> x \<in> U \<inter> W"
using \<open>openin X W\<close> \<open>x \<in> W\<close> \<open>openin X U\<close> \<open>x \<in> U\<close> by blast
then have "\<exists>u' v. openin X u' \<and> closedin X v \<and> x \<in> u' \<and> u' \<subseteq> v \<and> v \<subseteq> U \<and> v \<subseteq> W"
using reg
by (metis le_infE neighbourhood_base_of neighbourhood_base_of_closedin)
then show "\<exists>U V. openin X U \<and> compactin X V \<and> x \<in> U \<and> U \<subseteq> V \<and> V \<subseteq> W"
by (meson \<open>U \<subseteq> K\<close> \<open>compactin X K\<close> closed_compactin subset_trans)
qed
lemma Hausdorff_regular: "\<lbrakk>Hausdorff_space X; neighbourhood_base_of (compactin X) X\<rbrakk> \<Longrightarrow> regular_space X"
by (metis compactin_imp_closedin neighbourhood_base_of_closedin neighbourhood_base_of_mono)
lemma locally_compact_Hausdorff_imp_regular_space:
assumes loc: "locally_compact_space X" and "Hausdorff_space X"
shows "regular_space X"
unfolding neighbourhood_base_of_closedin [symmetric] neighbourhood_base_of
proof clarify
fix W x
assume "openin X W" and "x \<in> W"
then have "x \<in> topspace X"
using openin_subset by blast
then obtain U K where "openin X U" "compactin X K" and UK: "x \<in> U" "U \<subseteq> K"
by (meson loc locally_compact_space_def)
with \<open>Hausdorff_space X\<close> have "regular_space (subtopology X K)"
using Hausdorff_space_subtopology compact_Hausdorff_imp_regular_space compact_space_subtopology by blast
then have "\<exists>U' V'. openin (subtopology X K) U' \<and> closedin (subtopology X K) V' \<and> x \<in> U' \<and> U' \<subseteq> V' \<and> V' \<subseteq> K \<inter> W"
unfolding neighbourhood_base_of_closedin [symmetric] neighbourhood_base_of
by (meson IntI \<open>U \<subseteq> K\<close> \<open>openin X W\<close> \<open>x \<in> U\<close> \<open>x \<in> W\<close> openin_subtopology_Int2 subsetD)
then obtain V C where "openin X V" "closedin X C" and VC: "x \<in> K \<inter> V" "K \<inter> V \<subseteq> K \<inter> C" "K \<inter> C \<subseteq> K \<inter> W"
by (metis Int_commute closedin_subtopology openin_subtopology)
show "\<exists>U V. openin X U \<and> closedin X V \<and> x \<in> U \<and> U \<subseteq> V \<and> V \<subseteq> W"
proof (intro conjI exI)
show "openin X (U \<inter> V)"
using \<open>openin X U\<close> \<open>openin X V\<close> by blast
show "closedin X (K \<inter> C)"
using \<open>closedin X C\<close> \<open>compactin X K\<close> compactin_imp_closedin \<open>Hausdorff_space X\<close> by blast
qed (use UK VC in auto)
qed
lemma locally_compact_space_neighbourhood_base:
"Hausdorff_space X \<or> regular_space X
\<Longrightarrow> locally_compact_space X \<longleftrightarrow> neighbourhood_base_of (compactin X) X"
by (metis locally_compact_imp_neighbourhood_base locally_compact_Hausdorff_imp_regular_space
neighbourhood_base_imp_locally_compact_space)
lemma locally_compact_Hausdorff_or_regular:
"locally_compact_space X \<and> (Hausdorff_space X \<or> regular_space X) \<longleftrightarrow> locally_compact_space X \<and> regular_space X"
using locally_compact_Hausdorff_imp_regular_space by blast
lemma locally_compact_space_compact_closedin:
assumes "Hausdorff_space X \<or> regular_space X"
shows "locally_compact_space X \<longleftrightarrow>
(\<forall>x \<in> topspace X. \<exists>U K. openin X U \<and> compactin X K \<and> closedin X K \<and> x \<in> U \<and> U \<subseteq> K)"
using locally_compact_Hausdorff_or_regular unfolding locally_compact_space_def
by (metis assms closed_compactin inf.absorb_iff2 le_infE neighbourhood_base_of neighbourhood_base_of_closedin)
lemma locally_compact_space_compact_closure_of:
assumes "Hausdorff_space X \<or> regular_space X"
shows "locally_compact_space X \<longleftrightarrow>
(\<forall>x \<in> topspace X. \<exists>U. openin X U \<and> compactin X (X closure_of U) \<and> x \<in> U)" (is "?lhs=?rhs")
proof
assume ?lhs then show ?rhs
by (metis assms closed_compactin closedin_closure_of closure_of_eq closure_of_mono locally_compact_space_compact_closedin)
next
assume ?rhs then show ?lhs
by (meson closure_of_subset locally_compact_space_def openin_subset)
qed
lemma locally_compact_space_neighbourhood_base_closedin:
assumes "Hausdorff_space X \<or> regular_space X"
shows "locally_compact_space X \<longleftrightarrow> neighbourhood_base_of (\<lambda>C. compactin X C \<and> closedin X C) X" (is "?lhs=?rhs")
proof
assume L: ?lhs
then have "regular_space X"
using assms locally_compact_Hausdorff_imp_regular_space by blast
with L have "neighbourhood_base_of (compactin X) X"
by (simp add: locally_compact_imp_neighbourhood_base)
with \<open>regular_space X\<close> show ?rhs
by (smt (verit, ccfv_threshold) closed_compactin neighbourhood_base_of subset_trans neighbourhood_base_of_closedin)
next
assume ?rhs then show ?lhs
using neighbourhood_base_imp_locally_compact_space neighbourhood_base_of_mono by blast
qed
lemma locally_compact_space_neighbourhood_base_closure_of:
assumes "Hausdorff_space X \<or> regular_space X"
shows "locally_compact_space X \<longleftrightarrow> neighbourhood_base_of (\<lambda>T. compactin X (X closure_of T)) X"
(is "?lhs=?rhs")
proof
assume L: ?lhs
then have "regular_space X"
using assms locally_compact_Hausdorff_imp_regular_space by blast
with L have "neighbourhood_base_of (\<lambda>A. compactin X A \<and> closedin X A) X"
using locally_compact_space_neighbourhood_base_closedin by blast
then show ?rhs
by (simp add: closure_of_closedin neighbourhood_base_of_mono)
next
assume ?rhs then show ?lhs
unfolding locally_compact_space_def neighbourhood_base_of
by (meson closure_of_subset openin_topspace subset_trans)
qed
lemma locally_compact_space_neighbourhood_base_open_closure_of:
assumes "Hausdorff_space X \<or> regular_space X"
shows "locally_compact_space X \<longleftrightarrow>
neighbourhood_base_of (\<lambda>U. openin X U \<and> compactin X (X closure_of U)) X"
(is "?lhs=?rhs")
proof
assume L: ?lhs
then have "regular_space X"
using assms locally_compact_Hausdorff_imp_regular_space by blast
then have "neighbourhood_base_of (\<lambda>T. compactin X (X closure_of T)) X"
using L locally_compact_space_neighbourhood_base_closure_of by auto
with L show ?rhs
unfolding neighbourhood_base_of
by (meson closed_compactin closure_of_closure_of closure_of_eq closure_of_mono subset_trans)
next
assume ?rhs then show ?lhs
unfolding locally_compact_space_def neighbourhood_base_of
by (meson closure_of_subset openin_topspace subset_trans)
qed
lemma locally_compact_space_compact_closed_compact:
assumes "Hausdorff_space X \<or> regular_space X"
shows "locally_compact_space X \<longleftrightarrow>
(\<forall>K. compactin X K
\<longrightarrow> (\<exists>U L. openin X U \<and> compactin X L \<and> closedin X L \<and> K \<subseteq> U \<and> U \<subseteq> L))"
(is "?lhs=?rhs")
proof
assume L: ?lhs
then obtain U L where UL: "\<forall>x \<in> topspace X. openin X (U x) \<and> compactin X (L x) \<and> closedin X (L x) \<and> x \<in> U x \<and> U x \<subseteq> L x"
unfolding locally_compact_space_compact_closedin [OF assms]
by metis
show ?rhs
proof clarify
fix K
assume "compactin X K"
then have "K \<subseteq> topspace X"
by (simp add: compactin_subset_topspace)
then have *: "(\<forall>U\<in>U ` K. openin X U) \<and> K \<subseteq> \<Union> (U ` K)"
using UL by blast
with \<open>compactin X K\<close> obtain KF where KF: "finite KF" "KF \<subseteq> K" "K \<subseteq> \<Union>(U ` KF)"
by (metis compactinD finite_subset_image)
show "\<exists>U L. openin X U \<and> compactin X L \<and> closedin X L \<and> K \<subseteq> U \<and> U \<subseteq> L"
proof (intro conjI exI)
show "openin X (\<Union> (U ` KF))"
using "*" \<open>KF \<subseteq> K\<close> by fastforce
show "compactin X (\<Union> (L ` KF))"
by (smt (verit) UL \<open>K \<subseteq> topspace X\<close> KF compactin_Union finite_imageI imageE subset_iff)
show "closedin X (\<Union> (L ` KF))"
by (smt (verit) UL \<open>K \<subseteq> topspace X\<close> KF closedin_Union finite_imageI imageE subsetD)
qed (use UL \<open>K \<subseteq> topspace X\<close> KF in auto)
qed
next
assume ?rhs then show ?lhs
by (metis compactin_sing insert_subset locally_compact_space_def)
qed
lemma locally_compact_regular_space_neighbourhood_base:
"locally_compact_space X \<and> regular_space X \<longleftrightarrow>
neighbourhood_base_of (\<lambda>C. compactin X C \<and> closedin X C) X"
using locally_compact_space_neighbourhood_base_closedin neighbourhood_base_of_closedin neighbourhood_base_of_mono by blast
lemma locally_compact_kc_space:
"neighbourhood_base_of (compactin X) X \<and> kc_space X \<longleftrightarrow>
locally_compact_space X \<and> Hausdorff_space X"
using Hausdorff_imp_kc_space locally_compact_imp_kc_eq_Hausdorff_space locally_compact_space_neighbourhood_base by blast
lemma locally_compact_kc_space_alt:
"neighbourhood_base_of (compactin X) X \<and> kc_space X \<longleftrightarrow>
locally_compact_space X \<and> Hausdorff_space X \<and> regular_space X"
using Hausdorff_regular locally_compact_kc_space by blast
lemma locally_compact_kc_imp_regular_space:
"\<lbrakk>neighbourhood_base_of (compactin X) X; kc_space X\<rbrakk> \<Longrightarrow> regular_space X"
using Hausdorff_regular locally_compact_imp_kc_eq_Hausdorff_space by blast
lemma kc_locally_compact_space:
"kc_space X
\<Longrightarrow> neighbourhood_base_of (compactin X) X \<longleftrightarrow> locally_compact_space X \<and> Hausdorff_space X \<and> regular_space X"
using Hausdorff_regular locally_compact_kc_space by blast
lemma locally_compact_space_closed_subset:
assumes loc: "locally_compact_space X" and "closedin X S"
shows "locally_compact_space (subtopology X S)"
proof (clarsimp simp: locally_compact_space_def)
fix x assume x: "x \<in> topspace X" "x \<in> S"
then obtain U K where UK: "openin X U \<and> compactin X K \<and> x \<in> U \<and> U \<subseteq> K"
by (meson loc locally_compact_space_def)
show "\<exists>U. openin (subtopology X S) U \<and>
(\<exists>K. compactin (subtopology X S) K \<and> x \<in> U \<and> U \<subseteq> K)"
proof (intro conjI exI)
show "openin (subtopology X S) (S \<inter> U)"
by (simp add: UK openin_subtopology_Int2)
show "compactin (subtopology X S) (S \<inter> K)"
by (simp add: UK assms(2) closed_Int_compactin compactin_subtopology)
qed (use UK x in auto)
qed
lemma locally_compact_space_open_subset:
assumes X: "Hausdorff_space X \<or> regular_space X" and loc: "locally_compact_space X" and "openin X S"
shows "locally_compact_space (subtopology X S)"
proof (clarsimp simp: locally_compact_space_def)
fix x assume x: "x \<in> topspace X" "x \<in> S"
then obtain U K where UK: "openin X U" "compactin X K" "x \<in> U" "U \<subseteq> K"
by (meson loc locally_compact_space_def)
moreover have reg: "regular_space X"
using X loc locally_compact_Hausdorff_imp_regular_space by blast
moreover have "openin X (U \<inter> S)"
by (simp add: UK \<open>openin X S\<close> openin_Int)
ultimately obtain V C
where VC: "openin X V" "closedin X C" "x \<in> V" "V \<subseteq> C" "C \<subseteq> U" "C \<subseteq> S"
by (metis \<open>x \<in> S\<close> IntI le_inf_iff neighbourhood_base_of neighbourhood_base_of_closedin)
show "\<exists>U. openin (subtopology X S) U \<and>
(\<exists>K. compactin (subtopology X S) K \<and> x \<in> U \<and> U \<subseteq> K)"
proof (intro conjI exI)
show "openin (subtopology X S) V"
using VC by (meson \<open>openin X S\<close> openin_open_subtopology order_trans)
show "compactin (subtopology X S) (C \<inter> K)"
using UK VC closed_Int_compactin compactin_subtopology by fastforce
qed (use UK VC x in auto)
qed
lemma locally_compact_space_discrete_topology:
"locally_compact_space (discrete_topology U)"
by (simp add: neighbourhood_base_imp_locally_compact_space neighbourhood_base_of_discrete_topology)
lemma locally_compact_space_continuous_open_map_image:
"\<lbrakk>continuous_map X X' f; open_map X X' f;
f ` topspace X = topspace X'; locally_compact_space X\<rbrakk> \<Longrightarrow> locally_compact_space X'"
unfolding locally_compact_space_def open_map_def
by (smt (verit, ccfv_SIG) image_compactin image_iff image_mono)
lemma locally_compact_subspace_openin_closure_of:
assumes "Hausdorff_space X" and S: "S \<subseteq> topspace X"
and loc: "locally_compact_space (subtopology X S)"
shows "openin (subtopology X (X closure_of S)) S"
unfolding openin_subopen [where S=S]
proof clarify
fix a assume "a \<in> S"
then obtain T K where *: "openin X T" "compactin X K" "K \<subseteq> S" "a \<in> S" "a \<in> T" "S \<inter> T \<subseteq> K"
using loc unfolding locally_compact_space_def
by (metis IntE S compactin_subtopology inf_commute openin_subtopology topspace_subtopology_subset)
have "T \<inter> X closure_of S \<subseteq> X closure_of (T \<inter> S)"
by (simp add: "*"(1) openin_Int_closure_of_subset)
also have "... \<subseteq> S"
using * \<open>Hausdorff_space X\<close> by (metis closure_of_minimal compactin_imp_closedin order.trans inf_commute)
finally have "T \<inter> X closure_of S \<subseteq> T \<inter> S" by simp
then have "openin (subtopology X (X closure_of S)) (T \<inter> S)"
unfolding openin_subtopology using \<open>openin X T\<close> S closure_of_subset by fastforce
with * show "\<exists>T. openin (subtopology X (X closure_of S)) T \<and> a \<in> T \<and> T \<subseteq> S"
by blast
qed
lemma locally_compact_subspace_closed_Int_openin:
"\<lbrakk>Hausdorff_space X \<and> S \<subseteq> topspace X \<and> locally_compact_space(subtopology X S)\<rbrakk>
\<Longrightarrow> \<exists>C U. closedin X C \<and> openin X U \<and> C \<inter> U = S"
by (metis closedin_closure_of inf_commute locally_compact_subspace_openin_closure_of openin_subtopology)
lemma locally_compact_subspace_open_in_closure_of_eq:
assumes "Hausdorff_space X" and loc: "locally_compact_space X"
shows "openin (subtopology X (X closure_of S)) S \<longleftrightarrow> S \<subseteq> topspace X \<and> locally_compact_space(subtopology X S)" (is "?lhs=?rhs")
proof
assume L: ?lhs
then obtain "S \<subseteq> topspace X" "regular_space X"
using assms locally_compact_Hausdorff_imp_regular_space openin_subset by fastforce
then have "locally_compact_space (subtopology (subtopology X (X closure_of S)) S)"
by (simp add: L loc locally_compact_space_closed_subset locally_compact_space_open_subset regular_space_subtopology)
then show ?rhs
by (metis L inf.orderE inf_commute le_inf_iff openin_subset subtopology_subtopology topspace_subtopology)
next
assume ?rhs then show ?lhs
using assms locally_compact_subspace_openin_closure_of by blast
qed
lemma locally_compact_subspace_closed_Int_openin_eq:
assumes "Hausdorff_space X" and loc: "locally_compact_space X"
shows "(\<exists>C U. closedin X C \<and> openin X U \<and> C \<inter> U = S) \<longleftrightarrow> S \<subseteq> topspace X \<and> locally_compact_space(subtopology X S)" (is "?lhs=?rhs")
proof
assume L: ?lhs
then obtain C U where "closedin X C" "openin X U" and Seq: "S = C \<inter> U"
by blast
then have "C \<subseteq> topspace X"
by (simp add: closedin_subset)
have "locally_compact_space (subtopology (subtopology X C) (topspace (subtopology X C) \<inter> U))"
proof (rule locally_compact_space_open_subset)
show "locally_compact_space (subtopology X C)"
by (simp add: \<open>closedin X C\<close> loc locally_compact_space_closed_subset)
show "openin (subtopology X C) (topspace (subtopology X C) \<inter> U)"
by (simp add: \<open>openin X U\<close> Int_left_commute inf_commute openin_Int openin_subtopology_Int2)
qed (simp add: Hausdorff_space_subtopology \<open>Hausdorff_space X\<close>)
then show ?rhs
by (metis Seq \<open>C \<subseteq> topspace X\<close> inf.coboundedI1 subtopology_subtopology subtopology_topspace)
next
assume ?rhs then show ?lhs
using assms locally_compact_subspace_closed_Int_openin by blast
qed
lemma dense_locally_compact_openin_Hausdorff_space:
"\<lbrakk>Hausdorff_space X; S \<subseteq> topspace X; X closure_of S = topspace X;
locally_compact_space (subtopology X S)\<rbrakk> \<Longrightarrow> openin X S"
by (metis locally_compact_subspace_openin_closure_of subtopology_topspace)
lemma locally_compact_space_prod_topology:
"locally_compact_space (prod_topology X Y) \<longleftrightarrow>
(prod_topology X Y) = trivial_topology \<or>
locally_compact_space X \<and> locally_compact_space Y" (is "?lhs=?rhs")
proof (cases "(prod_topology X Y) = trivial_topology")
case True
then show ?thesis
using locally_compact_space_discrete_topology by force
next
case False
then obtain w z where wz: "w \<in> topspace X" "z \<in> topspace Y"
by fastforce
show ?thesis
proof
assume L: ?lhs then show ?rhs
by (metis locally_compact_space_retraction_map_image prod_topology_trivial_iff retraction_map_fst retraction_map_snd)
next
assume R: ?rhs
show ?lhs
unfolding locally_compact_space_def
proof clarsimp
fix x y
assume "x \<in> topspace X" and "y \<in> topspace Y"
obtain U C where "openin X U" "compactin X C" "x \<in> U" "U \<subseteq> C"
by (meson False R \<open>x \<in> topspace X\<close> locally_compact_space_def)
obtain V D where "openin Y V" "compactin Y D" "y \<in> V" "V \<subseteq> D"
by (meson False R \<open>y \<in> topspace Y\<close> locally_compact_space_def)
show "\<exists>U. openin (prod_topology X Y) U \<and> (\<exists>K. compactin (prod_topology X Y) K \<and> (x, y) \<in> U \<and> U \<subseteq> K)"
proof (intro exI conjI)
show "openin (prod_topology X Y) (U \<times> V)"
by (simp add: \<open>openin X U\<close> \<open>openin Y V\<close> openin_prod_Times_iff)
show "compactin (prod_topology X Y) (C \<times> D)"
by (simp add: \<open>compactin X C\<close> \<open>compactin Y D\<close> compactin_Times)
show "(x, y) \<in> U \<times> V"
by (simp add: \<open>x \<in> U\<close> \<open>y \<in> V\<close>)
show "U \<times> V \<subseteq> C \<times> D"
by (simp add: Sigma_mono \<open>U \<subseteq> C\<close> \<open>V \<subseteq> D\<close>)
qed
qed
qed
qed
lemma locally_compact_space_product_topology:
"locally_compact_space(product_topology X I) \<longleftrightarrow>
product_topology X I = trivial_topology \<or>
finite {i \<in> I. \<not> compact_space(X i)} \<and> (\<forall>i \<in> I. locally_compact_space(X i))" (is "?lhs=?rhs")
proof (cases "(product_topology X I) = trivial_topology")
case True
then show ?thesis
by (simp add: locally_compact_space_def)
next
case False
show ?thesis
proof
assume L: ?lhs
obtain z where z: "z \<in> topspace (product_topology X I)"
using False
by (meson ex_in_conv null_topspace_iff_trivial)
with L z obtain U C where "openin (product_topology X I) U" "compactin (product_topology X I) C" "z \<in> U" "U \<subseteq> C"
by (meson locally_compact_space_def)
then obtain V where finV: "finite {i \<in> I. V i \<noteq> topspace (X i)}" and "\<forall>i \<in> I. openin (X i) (V i)"
and "z \<in> PiE I V" "PiE I V \<subseteq> U"
by (auto simp: openin_product_topology_alt)
have "compact_space (X i)" if "i \<in> I" "V i = topspace (X i)" for i
proof -
have "compactin (X i) ((\<lambda>x. x i) ` C)"
using \<open>compactin (product_topology X I) C\<close> image_compactin
by (metis continuous_map_product_projection \<open>i \<in> I\<close>)
moreover have "V i \<subseteq> (\<lambda>x. x i) ` C"
proof -
have "V i \<subseteq> (\<lambda>x. x i) ` Pi\<^sub>E I V"
by (metis \<open>z \<in> Pi\<^sub>E I V\<close> empty_iff image_projection_PiE order_refl \<open>i \<in> I\<close>)
also have "\<dots> \<subseteq> (\<lambda>x. x i) ` C"
using \<open>U \<subseteq> C\<close> \<open>Pi\<^sub>E I V \<subseteq> U\<close> by blast
finally show ?thesis .
qed
ultimately show ?thesis
by (metis closed_compactin closedin_topspace compact_space_def that(2))
qed
with finV have "finite {i \<in> I. \<not> compact_space (X i)}"
by (metis (mono_tags, lifting) mem_Collect_eq finite_subset subsetI)
moreover have "locally_compact_space (X i)" if "i \<in> I" for i
by (meson False L locally_compact_space_retraction_map_image retraction_map_product_projection that)
ultimately show ?rhs by metis
next
assume R: ?rhs
show ?lhs
unfolding locally_compact_space_def
proof clarsimp
fix z
assume z: "z \<in> (\<Pi>\<^sub>E i\<in>I. topspace (X i))"
have "\<exists>U C. openin (X i) U \<and> compactin (X i) C \<and> z i \<in> U \<and> U \<subseteq> C \<and>
(compact_space(X i) \<longrightarrow> U = topspace(X i) \<and> C = topspace(X i))"
if "i \<in> I" for i
using that R z unfolding locally_compact_space_def compact_space_def
by (metis (no_types, lifting) False PiE_mem openin_topspace set_eq_subset)
then obtain U C where UC: "\<And>i. i \<in> I \<Longrightarrow>
openin (X i) (U i) \<and> compactin (X i) (C i) \<and> z i \<in> U i \<and> U i \<subseteq> C i \<and>
(compact_space(X i) \<longrightarrow> U i = topspace(X i) \<and> C i = topspace(X i))"
by metis
show "\<exists>U. openin (product_topology X I) U \<and> (\<exists>K. compactin (product_topology X I) K \<and> z \<in> U \<and> U \<subseteq> K)"
proof (intro exI conjI)
show "openin (product_topology X I) (Pi\<^sub>E I U)"
by (smt (verit) Collect_cong False R UC compactin_subspace openin_PiE_gen subset_antisym subtopology_topspace)
show "compactin (product_topology X I) (Pi\<^sub>E I C)"
by (simp add: UC compactin_PiE)
qed (use UC z in blast)+
qed
qed
qed
lemma locally_compact_space_sum_topology:
"locally_compact_space (sum_topology X I) \<longleftrightarrow> (\<forall>i \<in> I. locally_compact_space (X i))" (is "?lhs=?rhs")
proof
assume ?lhs then show ?rhs
by (metis closed_map_component_injection embedding_map_imp_homeomorphic_space embedding_map_component_injection
embedding_imp_closed_map_eq homeomorphic_locally_compact_space locally_compact_space_closed_subset)
next
assume R: ?rhs
show ?lhs
unfolding locally_compact_space_def
proof clarsimp
fix i y
assume "i \<in> I" and y: "y \<in> topspace (X i)"
then obtain U K where UK: "openin (X i) U" "compactin (X i) K" "y \<in> U" "U \<subseteq> K"
using R by (fastforce simp: locally_compact_space_def)
then show "\<exists>U. openin (sum_topology X I) U \<and> (\<exists>K. compactin (sum_topology X I) K \<and> (i, y) \<in> U \<and> U \<subseteq> K)"
by (metis \<open>i \<in> I\<close> continuous_map_component_injection image_compactin image_mono
imageI open_map_component_injection open_map_def)
qed
qed
lemma locally_compact_space_euclidean:
"locally_compact_space (euclidean::'a::heine_borel topology)"
unfolding locally_compact_space_def
proof (intro strip)
fix x::'a
assume "x \<in> topspace euclidean"
have "ball x 1 \<subseteq> cball x 1"
by auto
then show "\<exists>U K. openin euclidean U \<and> compactin euclidean K \<and> x \<in> U \<and> U \<subseteq> K"
by (metis Elementary_Metric_Spaces.open_ball centre_in_ball compact_cball compactin_euclidean_iff open_openin zero_less_one)
qed
lemma locally_compact_Euclidean_space:
"locally_compact_space(Euclidean_space n)"
using homeomorphic_locally_compact_space [OF homeomorphic_Euclidean_space_product_topology]
using locally_compact_space_product_topology locally_compact_space_euclidean by fastforce
proposition quotient_map_prod_right:
assumes loc: "locally_compact_space Z"
and reg: "Hausdorff_space Z \<or> regular_space Z"
and f: "quotient_map X Y f"
shows "quotient_map (prod_topology Z X) (prod_topology Z Y) (\<lambda>(x,y). (x,f y))"
proof -
define h where "h \<equiv> (\<lambda>(x::'a,y). (x,f y))"
have "continuous_map (prod_topology Z X) Y (f o snd)"
by (simp add: continuous_map_of_snd f quotient_imp_continuous_map)
then have cmh: "continuous_map (prod_topology Z X) (prod_topology Z Y) h"
by (simp add: h_def continuous_map_paired split_def continuous_map_fst o_def)
have fim: "f ` topspace X = topspace Y"
by (simp add: f quotient_imp_surjective_map)
moreover
have "openin (prod_topology Z X) {u \<in> topspace Z \<times> topspace X. h u \<in> W}
\<longleftrightarrow> openin (prod_topology Z Y) W" (is "?lhs=?rhs")
if W: "W \<subseteq> topspace Z \<times> topspace Y" for W
proof
define S where "S \<equiv> {u \<in> topspace Z \<times> topspace X. h u \<in> W}"
assume ?lhs
then have L: "openin (prod_topology Z X) S"
using S_def by blast
have "\<exists>T. openin (prod_topology Z Y) T \<and> (x0, z0) \<in> T \<and> T \<subseteq> W"
if \<section>: "(x0,z0) \<in> W" for x0 z0
proof -
have x0: "x0 \<in> topspace Z"
using W that by blast
obtain y0 where y0: "y0 \<in> topspace X" "f y0 = z0"
by (metis W fim imageE insert_absorb insert_subset mem_Sigma_iff \<section>)
then have "(x0, y0) \<in> S"
by (simp add: S_def h_def that x0)
have "continuous_map Z (prod_topology Z X) (\<lambda>x. (x, y0))"
by (simp add: continuous_map_paired y0)
with openin_continuous_map_preimage [OF _ L]
have ope_ZS: "openin Z {x \<in> topspace Z. (x,y0) \<in> S}"
by blast
obtain U U' where "openin Z U" "compactin Z U'" "closedin Z U'"
"x0 \<in> U" "U \<subseteq> U'" "U' \<subseteq> {x \<in> topspace Z. (x,y0) \<in> S}"
using loc ope_ZS x0 \<open>(x0, y0) \<in> S\<close>
by (force simp: locally_compact_space_neighbourhood_base_closedin [OF reg]
neighbourhood_base_of)
then have D: "U' \<times> {y0} \<subseteq> S"
by (auto simp: )
define V where "V \<equiv> {z \<in> topspace Y. U' \<times> {y \<in> topspace X. f y = z} \<subseteq> S}"
have "z0 \<in> V"
using D y0 Int_Collect fim by (fastforce simp: h_def V_def S_def)
have "openin X {x \<in> topspace X. f x \<in> V} \<Longrightarrow> openin Y V"
using f unfolding V_def quotient_map_def subset_iff
by (smt (verit, del_insts) Collect_cong mem_Collect_eq)
moreover have "openin X {x \<in> topspace X. f x \<in> V}"
proof -
let ?Z = "subtopology Z U'"
have *: "{x \<in> topspace X. f x \<in> V} = topspace X - snd ` (U' \<times> topspace X - S)"
by (force simp: V_def S_def h_def simp flip: fim)
have "compact_space ?Z"
using \<open>compactin Z U'\<close> compactin_subspace by auto
moreover have "closedin (prod_topology ?Z X) (U' \<times> topspace X - S)"
by (simp add: L \<open>closedin Z U'\<close> closedin_closed_subtopology closedin_diff closedin_prod_Times_iff
prod_topology_subtopology(1))
ultimately show ?thesis
using "*" closed_map_snd closed_map_def by fastforce
qed
ultimately have "openin Y V"
by metis
show ?thesis
proof (intro conjI exI)
show "openin (prod_topology Z Y) (U \<times> V)"
by (simp add: openin_prod_Times_iff \<open>openin Z U\<close> \<open>openin Y V\<close>)
show "(x0, z0) \<in> U \<times> V"
by (simp add: \<open>x0 \<in> U\<close> \<open>z0 \<in> V\<close>)
show "U \<times> V \<subseteq> W"
using \<open>U \<subseteq> U'\<close> by (force simp: V_def S_def h_def simp flip: fim)
qed
qed
with openin_subopen show ?rhs by force
next
assume ?rhs then show ?lhs
using openin_continuous_map_preimage cmh by fastforce
qed
ultimately show ?thesis
by (fastforce simp: image_iff quotient_map_def h_def)
qed
lemma quotient_map_prod_left:
assumes loc: "locally_compact_space Z"
and reg: "Hausdorff_space Z \<or> regular_space Z"
and f: "quotient_map X Y f"
shows "quotient_map (prod_topology X Z) (prod_topology Y Z) (\<lambda>(x,y). (f x,y))"
proof -
have "(\<lambda>(x,y). (f x,y)) = prod.swap \<circ> (\<lambda>(x,y). (x,f y)) \<circ> prod.swap"
by force
then
show ?thesis
apply (rule ssubst)
proof (intro quotient_map_compose)
show "quotient_map (prod_topology X Z) (prod_topology Z X) prod.swap"
"quotient_map (prod_topology Z Y) (prod_topology Y Z) prod.swap"
using homeomorphic_map_def homeomorphic_map_swap quotient_map_eq by fastforce+
show "quotient_map (prod_topology Z X) (prod_topology Z Y) (\<lambda>(x, y). (x, f y))"
by (simp add: f loc quotient_map_prod_right reg)
qed
qed
lemma locally_compact_space_perfect_map_preimage:
assumes "locally_compact_space X'" and f: "perfect_map X X' f"
shows "locally_compact_space X"
unfolding locally_compact_space_def
proof (intro strip)
fix x
assume x: "x \<in> topspace X"
then obtain U K where "openin X' U" "compactin X' K" "f x \<in> U" "U \<subseteq> K"
using assms unfolding locally_compact_space_def perfect_map_def
by (metis (no_types, lifting) continuous_map_closedin Pi_iff)
show "\<exists>U K. openin X U \<and> compactin X K \<and> x \<in> U \<and> U \<subseteq> K"
proof (intro exI conjI)
have "continuous_map X X' f"
using f perfect_map_def by blast
then show "openin X {x \<in> topspace X. f x \<in> U}"
by (simp add: \<open>openin X' U\<close> continuous_map)
show "compactin X {x \<in> topspace X. f x \<in> K}"
using \<open>compactin X' K\<close> f perfect_imp_proper_map proper_map_alt by blast
qed (use x \<open>f x \<in> U\<close> \<open>U \<subseteq> K\<close> in auto)
qed
subsection\<open>Special characterizations of classes of functions into and out of R\<close>
lemma monotone_map_into_euclideanreal_alt:
assumes "continuous_map X euclideanreal f"
shows "(\<forall>k. is_interval k \<longrightarrow> connectedin X {x \<in> topspace X. f x \<in> k}) \<longleftrightarrow>
connected_space X \<and> monotone_map X euclideanreal f" (is "?lhs=?rhs")
proof
assume L: ?lhs
show ?rhs
proof
show "connected_space X"
using L connected_space_subconnected by blast
have "connectedin X {x \<in> topspace X. f x \<in> {y}}" for y
by (metis L is_interval_1 nle_le singletonD)
then show "monotone_map X euclideanreal f"
by (simp add: monotone_map)
qed
next
assume R: ?rhs
then
have *: False
if "a < b" "closedin X U" "closedin X V" "U \<noteq> {}" "V \<noteq> {}" "disjnt U V"
and UV: "{x \<in> topspace X. f x \<in> {a..b}} = U \<union> V"
and dis: "disjnt U {x \<in> topspace X. f x = b}" "disjnt V {x \<in> topspace X. f x = a}"
for a b U V
proof -
define E1 where "E1 \<equiv> U \<union> {x \<in> topspace X. f x \<in> {c. c \<le> a}}"
define E2 where "E2 \<equiv> V \<union> {x \<in> topspace X. f x \<in> {c. b \<le> c}}"
have "closedin X {x \<in> topspace X. f x \<le> a}" "closedin X {x \<in> topspace X. b \<le> f x}"
using assms continuous_map_upper_lower_semicontinuous_le by blast+
then have "closedin X E1" "closedin X E2"
unfolding E1_def E2_def using that by auto
moreover
have "E1 \<inter> E2 = {}"
unfolding E1_def E2_def using \<open>a<b\<close> \<open>disjnt U V\<close> dis UV
by (simp add: disjnt_def set_eq_iff) (smt (verit))
have "topspace X \<subseteq> E1 \<union> E2"
unfolding E1_def E2_def using UV by fastforce
have "E1 = {} \<or> E2 = {}"
using R connected_space_closedin
using \<open>E1 \<inter> E2 = {}\<close> \<open>closedin X E1\<close> \<open>closedin X E2\<close> \<open>topspace X \<subseteq> E1 \<union> E2\<close> by blast
then show False
using E1_def E2_def \<open>U \<noteq> {}\<close> \<open>V \<noteq> {}\<close> by fastforce
qed
show ?lhs
proof (intro strip)
fix K :: "real set"
assume "is_interval K"
have False
if "a \<in> K" "b \<in> K" and clo: "closedin X U" "closedin X V"
and UV: "{x. x \<in> topspace X \<and> f x \<in> K} \<subseteq> U \<union> V"
"U \<inter> V \<inter> {x. x \<in> topspace X \<and> f x \<in> K} = {}"
and nondis: "\<not> disjnt U {x. x \<in> topspace X \<and> f x = a}"
"\<not> disjnt V {x. x \<in> topspace X \<and> f x = b}"
for a b U V
proof -
have "\<forall>y. connectedin X {x. x \<in> topspace X \<and> f x = y}"
using R monotone_map by fastforce
then have **: False if "p \<in> U \<and> q \<in> V \<and> f p = f q \<and> f q \<in> K" for p q
unfolding connectedin_closedin
using \<open>a \<in> K\<close> \<open>b \<in> K\<close> UV clo that
by (smt (verit, ccfv_threshold) closedin_subset disjoint_iff mem_Collect_eq subset_iff)
consider "a < b" | "a = b" | "b < a"
by linarith
then show ?thesis
proof cases
case 1
define W where "W \<equiv> {x \<in> topspace X. f x \<in> {a..b}}"
have "closedin X W"
unfolding W_def
by (metis (no_types) assms closed_real_atLeastAtMost closed_closedin continuous_map_closedin)
show ?thesis
proof (rule * [OF 1 , of "U \<inter> W" "V \<inter> W"])
show "closedin X (U \<inter> W)" "closedin X (V \<inter> W)"
using \<open>closedin X W\<close> clo by auto
show "U \<inter> W \<noteq> {}" "V \<inter> W \<noteq> {}"
using nondis 1 by (auto simp: disjnt_iff W_def)
show "disjnt (U \<inter> W) (V \<inter> W)"
using \<open>is_interval K\<close> unfolding is_interval_1 disjnt_iff W_def
by (metis (mono_tags, lifting) \<open>a \<in> K\<close> \<open>b \<in> K\<close> ** Int_Collect atLeastAtMost_iff)
have "\<And>x. \<lbrakk>x \<in> topspace X; a \<le> f x; f x \<le> b\<rbrakk> \<Longrightarrow> x \<in> U \<or> x \<in> V"
using \<open>a \<in> K\<close> \<open>b \<in> K\<close> \<open>is_interval K\<close> UV unfolding is_interval_1 disjnt_iff
by blast
then show "{x \<in> topspace X. f x \<in> {a..b}} = U \<inter> W \<union> V \<inter> W"
by (auto simp: W_def)
show "disjnt (U \<inter> W) {x \<in> topspace X. f x = b}" "disjnt (V \<inter> W) {x \<in> topspace X. f x = a}"
using ** \<open>a \<in> K\<close> \<open>b \<in> K\<close> nondis by (force simp: disjnt_iff)+
qed
next
case 2
then show ?thesis
using ** nondis \<open>b \<in> K\<close> by (force simp add: disjnt_iff)
next
case 3
define W where "W \<equiv> {x \<in> topspace X. f x \<in> {b..a}}"
have "closedin X W"
unfolding W_def
by (metis (no_types) assms closed_real_atLeastAtMost closed_closedin continuous_map_closedin)
show ?thesis
proof (rule * [OF 3, of "V \<inter> W" "U \<inter> W"])
show "closedin X (U \<inter> W)" "closedin X (V \<inter> W)"
using \<open>closedin X W\<close> clo by auto
show "U \<inter> W \<noteq> {}" "V \<inter> W \<noteq> {}"
using nondis 3 by (auto simp: disjnt_iff W_def)
show "disjnt (V \<inter> W) (U \<inter> W)"
using \<open>is_interval K\<close> unfolding is_interval_1 disjnt_iff W_def
by (metis (mono_tags, lifting) \<open>a \<in> K\<close> \<open>b \<in> K\<close> ** Int_Collect atLeastAtMost_iff)
have "\<And>x. \<lbrakk>x \<in> topspace X; b \<le> f x; f x \<le> a\<rbrakk> \<Longrightarrow> x \<in> U \<or> x \<in> V"
using \<open>a \<in> K\<close> \<open>b \<in> K\<close> \<open>is_interval K\<close> UV unfolding is_interval_1 disjnt_iff
by blast
then show "{x \<in> topspace X. f x \<in> {b..a}} = V \<inter> W \<union> U \<inter> W"
by (auto simp: W_def)
show "disjnt (V \<inter> W) {x \<in> topspace X. f x = a}" "disjnt (U \<inter> W) {x \<in> topspace X. f x = b}"
using ** \<open>a \<in> K\<close> \<open>b \<in> K\<close> nondis by (force simp: disjnt_iff)+
qed
qed
qed
then show "connectedin X {x \<in> topspace X. f x \<in> K}"
unfolding connectedin_closedin disjnt_iff by blast
qed
qed
lemma monotone_map_into_euclideanreal:
"\<lbrakk>connected_space X; continuous_map X euclideanreal f\<rbrakk>
\<Longrightarrow> monotone_map X euclideanreal f \<longleftrightarrow>
(\<forall>k. is_interval k \<longrightarrow> connectedin X {x \<in> topspace X. f x \<in> k})"
by (simp add: monotone_map_into_euclideanreal_alt)
lemma monotone_map_euclideanreal_alt:
"(\<forall>I::real set. is_interval I \<longrightarrow> is_interval {x::real. x \<in> S \<and> f x \<in> I}) \<longleftrightarrow>
is_interval S \<and> (mono_on S f \<or> antimono_on S f)" (is "?lhs=?rhs")
proof
assume L [rule_format]: ?lhs
show ?rhs
proof
show "is_interval S"
using L is_interval_1 by auto
have False if "a \<in> S" "b \<in> S" "c \<in> S" "a<b" "b<c" and d: "f a < f b \<and> f c < f b \<or> f a > f b \<and> f c > f b" for a b c
using d
proof
assume "f a < f b \<and> f c < f b"
then show False
using L [of "{y. y < f b}"] unfolding is_interval_1
by (smt (verit, best) mem_Collect_eq that)
next
assume "f b < f a \<and> f b < f c"
then show False
using L [of "{y. y > f b}"] unfolding is_interval_1
by (smt (verit, best) mem_Collect_eq that)
qed
then show "mono_on S f \<or> monotone_on S (\<le>) (\<ge>) f"
unfolding monotone_on_def by (smt (verit))
qed
next
assume ?rhs then show ?lhs
unfolding is_interval_1 monotone_on_def by simp meson
qed
lemma monotone_map_euclideanreal:
fixes S :: "real set"
shows
"\<lbrakk>is_interval S; continuous_on S f\<rbrakk> \<Longrightarrow>
monotone_map (top_of_set S) euclideanreal f \<longleftrightarrow> (mono_on S f \<or> monotone_on S (\<le>) (\<ge>) f)"
using monotone_map_euclideanreal_alt
by (simp add: monotone_map_into_euclideanreal connectedin_subtopology is_interval_connected_1)
lemma injective_eq_monotone_map:
fixes f :: "real \<Rightarrow> real"
assumes "is_interval S" "continuous_on S f"
shows "inj_on f S \<longleftrightarrow> strict_mono_on S f \<or> strict_antimono_on S f"
by (metis assms injective_imp_monotone_map monotone_map_euclideanreal strict_antimono_iff_antimono
strict_mono_iff_mono top_greatest topspace_euclidean topspace_euclidean_subtopology)
subsection\<open>Normal spaces\<close>
definition normal_space
where "normal_space X \<equiv>
\<forall>S T. closedin X S \<and> closedin X T \<and> disjnt S T
\<longrightarrow> (\<exists>U V. openin X U \<and> openin X V \<and> S \<subseteq> U \<and> T \<subseteq> V \<and> disjnt U V)"
lemma normal_space_retraction_map_image:
assumes r: "retraction_map X Y r" and X: "normal_space X"
shows "normal_space Y"
unfolding normal_space_def
proof clarify
fix S T
assume "closedin Y S" and "closedin Y T" and "disjnt S T"
obtain r' where r': "retraction_maps X Y r r'"
using r retraction_map_def by blast
have "closedin X {x \<in> topspace X. r x \<in> S}" "closedin X {x \<in> topspace X. r x \<in> T}"
using closedin_continuous_map_preimage \<open>closedin Y S\<close> \<open>closedin Y T\<close> r'
by (auto simp: retraction_maps_def)
moreover
have "disjnt {x \<in> topspace X. r x \<in> S} {x \<in> topspace X. r x \<in> T}"
using \<open>disjnt S T\<close> by (auto simp: disjnt_def)
ultimately
obtain U V where UV: "openin X U \<and> openin X V \<and> {x \<in> topspace X. r x \<in> S} \<subseteq> U \<and> {x \<in> topspace X. r x \<in> T} \<subseteq> V" "disjnt U V"
by (meson X normal_space_def)
show "\<exists>U V. openin Y U \<and> openin Y V \<and> S \<subseteq> U \<and> T \<subseteq> V \<and> disjnt U V"
proof (intro exI conjI)
show "openin Y {x \<in> topspace Y. r' x \<in> U}" "openin Y {x \<in> topspace Y. r' x \<in> V}"
using openin_continuous_map_preimage UV r'
by (auto simp: retraction_maps_def)
show "S \<subseteq> {x \<in> topspace Y. r' x \<in> U}" "T \<subseteq> {x \<in> topspace Y. r' x \<in> V}"
using openin_continuous_map_preimage UV r' \<open>closedin Y S\<close> \<open>closedin Y T\<close>
by (auto simp add: closedin_def continuous_map_closedin retraction_maps_def subset_iff Pi_iff)
show "disjnt {x \<in> topspace Y. r' x \<in> U} {x \<in> topspace Y. r' x \<in> V}"
using \<open>disjnt U V\<close> by (auto simp: disjnt_def)
qed
qed
lemma homeomorphic_normal_space:
"X homeomorphic_space Y \<Longrightarrow> normal_space X \<longleftrightarrow> normal_space Y"
unfolding homeomorphic_space_def
by (meson homeomorphic_imp_retraction_maps homeomorphic_maps_sym normal_space_retraction_map_image retraction_map_def)
lemma normal_space:
"normal_space X \<longleftrightarrow>
(\<forall>S T. closedin X S \<and> closedin X T \<and> disjnt S T
\<longrightarrow> (\<exists>U. openin X U \<and> S \<subseteq> U \<and> disjnt T (X closure_of U)))"
proof -
have "(\<exists>V. openin X U \<and> openin X V \<and> S \<subseteq> U \<and> T \<subseteq> V \<and> disjnt U V) \<longleftrightarrow> openin X U \<and> S \<subseteq> U \<and> disjnt T (X closure_of U)"
(is "?lhs=?rhs")
if "closedin X S" "closedin X T" "disjnt S T" for S T U
proof
show "?lhs \<Longrightarrow> ?rhs"
by (smt (verit, best) disjnt_iff in_closure_of subsetD)
assume R: ?rhs
then have "(U \<union> S) \<inter> (topspace X - X closure_of U) = {}"
by (metis Diff_eq_empty_iff Int_Diff Int_Un_eq(4) closure_of_subset inf.orderE openin_subset)
moreover have "T \<subseteq> topspace X - X closure_of U"
by (meson DiffI R closedin_subset disjnt_iff subsetD subsetI that(2))
ultimately show ?lhs
by (metis R closedin_closure_of closedin_def disjnt_def sup.orderE)
qed
then show ?thesis
unfolding normal_space_def by meson
qed
lemma normal_space_alt:
"normal_space X \<longleftrightarrow>
(\<forall>S U. closedin X S \<and> openin X U \<and> S \<subseteq> U \<longrightarrow> (\<exists>V. openin X V \<and> S \<subseteq> V \<and> X closure_of V \<subseteq> U))"
proof -
have "\<exists>V. openin X V \<and> S \<subseteq> V \<and> X closure_of V \<subseteq> U"
if "\<And>T. closedin X T \<longrightarrow> disjnt S T \<longrightarrow> (\<exists>U. openin X U \<and> S \<subseteq> U \<and> disjnt T (X closure_of U))"
"closedin X S" "openin X U" "S \<subseteq> U"
for S U
using that
by (smt (verit) Diff_eq_empty_iff Int_Diff closure_of_subset_topspace disjnt_def inf.orderE inf_commute openin_closedin_eq)
moreover have "\<exists>U. openin X U \<and> S \<subseteq> U \<and> disjnt T (X closure_of U)"
if "\<And>U. openin X U \<and> S \<subseteq> U \<longrightarrow> (\<exists>V. openin X V \<and> S \<subseteq> V \<and> X closure_of V \<subseteq> U)"
and "closedin X S" "closedin X T" "disjnt S T"
for S T
using that
by (smt (verit) Diff_Diff_Int Diff_eq_empty_iff Int_Diff closedin_def disjnt_def inf.absorb_iff2 inf.orderE)
ultimately show ?thesis
by (fastforce simp: normal_space)
qed
lemma normal_space_closures:
"normal_space X \<longleftrightarrow>
(\<forall>S T. S \<subseteq> topspace X \<and> T \<subseteq> topspace X \<and>
disjnt (X closure_of S) (X closure_of T)
\<longrightarrow> (\<exists>U V. openin X U \<and> openin X V \<and> S \<subseteq> U \<and> T \<subseteq> V \<and> disjnt U V))"
(is "?lhs=?rhs")
proof
show "?lhs \<Longrightarrow> ?rhs"
by (meson closedin_closure_of closure_of_subset normal_space_def order.trans)
show "?rhs \<Longrightarrow> ?lhs"
by (metis closedin_subset closure_of_eq normal_space_def)
qed
lemma normal_space_disjoint_closures:
"normal_space X \<longleftrightarrow>
(\<forall>S T. closedin X S \<and> closedin X T \<and> disjnt S T
\<longrightarrow> (\<exists>U V. openin X U \<and> openin X V \<and> S \<subseteq> U \<and> T \<subseteq> V \<and>
disjnt (X closure_of U) (X closure_of V)))"
(is "?lhs=?rhs")
proof
show "?lhs \<Longrightarrow> ?rhs"
by (metis closedin_closure_of normal_space)
show "?rhs \<Longrightarrow> ?lhs"
by (smt (verit) closure_of_subset disjnt_iff normal_space openin_subset subset_eq)
qed
lemma normal_space_dual:
"normal_space X \<longleftrightarrow>
(\<forall>U V. openin X U \<longrightarrow> openin X V \<and> U \<union> V = topspace X
\<longrightarrow> (\<exists>S T. closedin X S \<and> closedin X T \<and> S \<subseteq> U \<and> T \<subseteq> V \<and> S \<union> T = topspace X))"
(is "_ = ?rhs")
proof -
have "normal_space X \<longleftrightarrow>
(\<forall>U V. closedin X U \<longrightarrow> closedin X V \<longrightarrow> disjnt U V \<longrightarrow>
(\<exists>S T. \<not> (openin X S \<and> openin X T \<longrightarrow>
\<not> (U \<subseteq> S \<and> V \<subseteq> T \<and> disjnt S T))))"
unfolding normal_space_def by meson
also have "... \<longleftrightarrow> (\<forall>U V. openin X U \<longrightarrow> openin X V \<and> disjnt (topspace X - U) (topspace X - V) \<longrightarrow>
(\<exists>S T. \<not> (openin X S \<and> openin X T \<longrightarrow>
\<not> (topspace X - U \<subseteq> S \<and> topspace X - V \<subseteq> T \<and> disjnt S T))))"
by (auto simp: all_closedin)
also have "... \<longleftrightarrow> ?rhs"
proof -
have *: "disjnt (topspace X - U) (topspace X - V) \<longleftrightarrow> U \<union> V = topspace X"
if "U \<subseteq> topspace X" "V \<subseteq> topspace X" for U V
using that by (auto simp: disjnt_iff)
show ?thesis
using ex_closedin *
apply (simp add: ex_closedin * [OF openin_subset openin_subset] cong: conj_cong)
apply (intro all_cong1 ex_cong1 imp_cong refl)
by (smt (verit, best) "*" Diff_Diff_Int Diff_subset Diff_subset_conv inf.orderE inf_commute openin_subset sup_commute)
qed
finally show ?thesis .
qed
lemma normal_t1_imp_Hausdorff_space:
assumes "normal_space X" "t1_space X"
shows "Hausdorff_space X"
unfolding Hausdorff_space_def
proof clarify
fix x y
assume xy: "x \<in> topspace X" "y \<in> topspace X" "x \<noteq> y"
then have "disjnt {x} {y}"
by (auto simp: disjnt_iff)
then show "\<exists>U V. openin X U \<and> openin X V \<and> x \<in> U \<and> y \<in> V \<and> disjnt U V"
using assms xy closedin_t1_singleton normal_space_def
by (metis singletonI subsetD)
qed
lemma normal_t1_eq_Hausdorff_space:
"normal_space X \<Longrightarrow> t1_space X \<longleftrightarrow> Hausdorff_space X"
using normal_t1_imp_Hausdorff_space t1_or_Hausdorff_space by blast
lemma normal_t1_imp_regular_space:
"\<lbrakk>normal_space X; t1_space X\<rbrakk> \<Longrightarrow> regular_space X"
by (metis compactin_imp_closedin normal_space_def normal_t1_eq_Hausdorff_space regular_space_compact_closed_sets)
lemma compact_Hausdorff_or_regular_imp_normal_space:
"\<lbrakk>compact_space X; Hausdorff_space X \<or> regular_space X\<rbrakk>
\<Longrightarrow> normal_space X"
by (metis Hausdorff_space_compact_sets closedin_compact_space normal_space_def regular_space_compact_closed_sets)
lemma normal_space_discrete_topology:
"normal_space(discrete_topology U)"
by (metis discrete_topology_closure_of inf_le2 normal_space_alt)
lemma normal_space_fsigmas:
"normal_space X \<longleftrightarrow>
(\<forall>S T. fsigma_in X S \<and> fsigma_in X T \<and> separatedin X S T
\<longrightarrow> (\<exists>U B. openin X U \<and> openin X B \<and> S \<subseteq> U \<and> T \<subseteq> B \<and> disjnt U B))" (is "?lhs=?rhs")
proof
assume L: ?lhs
show ?rhs
proof clarify
fix S T
assume "fsigma_in X S"
then obtain C where C: "\<And>n. closedin X (C n)" "\<And>n. C n \<subseteq> C (Suc n)" "\<Union> (range C) = S"
by (meson fsigma_in_ascending)
assume "fsigma_in X T"
then obtain D where D: "\<And>n. closedin X (D n)" "\<And>n. D n \<subseteq> D (Suc n)" "\<Union> (range D) = T"
by (meson fsigma_in_ascending)
assume "separatedin X S T"
have "\<And>n. disjnt (D n) (X closure_of S)"
by (metis D(3) \<open>separatedin X S T\<close> disjnt_Union1 disjnt_def rangeI separatedin_def)
then have "\<And>n. \<exists>V V'. openin X V \<and> openin X V' \<and> D n \<subseteq> V \<and> X closure_of S \<subseteq> V' \<and> disjnt V V'"
by (metis D(1) L closedin_closure_of normal_space_def)
then obtain V V' where V: "\<And>n. openin X (V n)" and "\<And>n. openin X (V' n)" "\<And>n. disjnt (V n) (V' n)"
and DV: "\<And>n. D n \<subseteq> V n"
and subV': "\<And>n. X closure_of S \<subseteq> V' n"
by metis
then have VV: "V' n \<inter> X closure_of V n = {}" for n
using openin_Int_closure_of_eq_empty [of X "V' n" "V n"] by (simp add: Int_commute disjnt_def)
have "\<And>n. disjnt (C n) (X closure_of T)"
by (metis C(3) \<open>separatedin X S T\<close> disjnt_Union1 disjnt_def rangeI separatedin_def)
then have "\<And>n. \<exists>U U'. openin X U \<and> openin X U' \<and> C n \<subseteq> U \<and> X closure_of T \<subseteq> U' \<and> disjnt U U'"
by (metis C(1) L closedin_closure_of normal_space_def)
then obtain U U' where U: "\<And>n. openin X (U n)" and "\<And>n. openin X (U' n)" "\<And>n. disjnt (U n) (U' n)"
and CU: "\<And>n. C n \<subseteq> U n"
and subU': "\<And>n. X closure_of T \<subseteq> U' n"
by metis
then have UU: "U' n \<inter> X closure_of U n = {}" for n
using openin_Int_closure_of_eq_empty [of X "U' n" "U n"] by (simp add: Int_commute disjnt_def)
show "\<exists>U B. openin X U \<and> openin X B \<and> S \<subseteq> U \<and> T \<subseteq> B \<and> disjnt U B"
proof (intro conjI exI)
have "\<And>S n. closedin X (\<Union>m\<le>n. X closure_of V m)"
by (force intro: closedin_Union)
then show "openin X (\<Union>n. U n - (\<Union>m\<le>n. X closure_of V m))"
using U by blast
have "\<And>S n. closedin X (\<Union>m\<le>n. X closure_of U m)"
by (force intro: closedin_Union)
then show "openin X (\<Union>n. V n - (\<Union>m\<le>n. X closure_of U m))"
using V by blast
have "S \<subseteq> topspace X"
by (simp add: \<open>fsigma_in X S\<close> fsigma_in_subset)
then show "S \<subseteq> (\<Union>n. U n - (\<Union>m\<le>n. X closure_of V m))"
apply (clarsimp simp: Ball_def)
by (metis VV C(3) CU IntI UN_E closure_of_subset empty_iff subV' subsetD)
have "T \<subseteq> topspace X"
by (simp add: \<open>fsigma_in X T\<close> fsigma_in_subset)
then show "T \<subseteq> (\<Union>n. V n - (\<Union>m\<le>n. X closure_of U m))"
apply (clarsimp simp: Ball_def)
by (metis UU D(3) DV IntI UN_E closure_of_subset empty_iff subU' subsetD)
have "\<And>x m n. \<lbrakk>x \<in> U n; x \<in> V m; \<forall>k\<le>m. x \<notin> X closure_of U k\<rbrakk> \<Longrightarrow> \<exists>k\<le>n. x \<in> X closure_of V k"
by (meson U V closure_of_subset nat_le_linear openin_subset subsetD)
then show "disjnt (\<Union>n. U n - (\<Union>m\<le>n. X closure_of V m)) (\<Union>n. V n - (\<Union>m\<le>n. X closure_of U m))"
by (force simp: disjnt_iff)
qed
qed
next
show "?rhs \<Longrightarrow> ?lhs"
by (simp add: closed_imp_fsigma_in normal_space_def separatedin_closed_sets)
qed
lemma normal_space_fsigma_subtopology:
assumes "normal_space X" "fsigma_in X S"
shows "normal_space (subtopology X S)"
unfolding normal_space_fsigmas
proof clarify
fix T U
assume "fsigma_in (subtopology X S) T"
and "fsigma_in (subtopology X S) U"
and TU: "separatedin (subtopology X S) T U"
then obtain A B where "openin X A \<and> openin X B \<and> T \<subseteq> A \<and> U \<subseteq> B \<and> disjnt A B"
by (metis assms fsigma_in_fsigma_subtopology normal_space_fsigmas separatedin_subtopology)
then
show "\<exists>A B. openin (subtopology X S) A \<and> openin (subtopology X S) B \<and> T \<subseteq> A \<and>
U \<subseteq> B \<and> disjnt A B"
using TU
by (force simp add: separatedin_subtopology openin_subtopology_alt disjnt_iff)
qed
lemma normal_space_closed_subtopology:
assumes "normal_space X" "closedin X S"
shows "normal_space (subtopology X S)"
by (simp add: assms closed_imp_fsigma_in normal_space_fsigma_subtopology)
lemma normal_space_continuous_closed_map_image:
assumes "normal_space X" and contf: "continuous_map X Y f"
and clof: "closed_map X Y f" and fim: "f ` topspace X = topspace Y"
shows "normal_space Y"
unfolding normal_space_def
proof clarify
fix S T
assume "closedin Y S" and "closedin Y T" and "disjnt S T"
have "closedin X {x \<in> topspace X. f x \<in> S}" "closedin X {x \<in> topspace X. f x \<in> T}"
using \<open>closedin Y S\<close> \<open>closedin Y T\<close> closedin_continuous_map_preimage contf by auto
moreover
have "disjnt {x \<in> topspace X. f x \<in> S} {x \<in> topspace X. f x \<in> T}"
using \<open>disjnt S T\<close> by (auto simp: disjnt_iff)
ultimately
obtain U V where "closedin X U" "closedin X V"
and subXU: "{x \<in> topspace X. f x \<in> S} \<subseteq> topspace X - U"
and subXV: "{x \<in> topspace X. f x \<in> T} \<subseteq> topspace X - V"
and dis: "disjnt (topspace X - U) (topspace X -V)"
using \<open>normal_space X\<close> by (force simp add: normal_space_def ex_openin)
have "closedin Y (f ` U)" "closedin Y (f ` V)"
using \<open>closedin X U\<close> \<open>closedin X V\<close> clof closed_map_def by blast+
moreover have "S \<subseteq> topspace Y - f ` U"
using \<open>closedin Y S\<close> \<open>closedin X U\<close> subXU by (force dest: closedin_subset)
moreover have "T \<subseteq> topspace Y - f ` V"
using \<open>closedin Y T\<close> \<open>closedin X V\<close> subXV by (force dest: closedin_subset)
moreover have "disjnt (topspace Y - f ` U) (topspace Y - f ` V)"
using fim dis by (force simp add: disjnt_iff)
ultimately show "\<exists>U V. openin Y U \<and> openin Y V \<and> S \<subseteq> U \<and> T \<subseteq> V \<and> disjnt U V"
by (force simp add: ex_openin)
qed
subsection \<open>Hereditary topological properties\<close>
definition hereditarily
where "hereditarily P X \<equiv>
\<forall>S. S \<subseteq> topspace X \<longrightarrow> P(subtopology X S)"
lemma hereditarily:
"hereditarily P X \<longleftrightarrow> (\<forall>S. P(subtopology X S))"
by (metis Int_lower1 hereditarily_def subtopology_restrict)
lemma hereditarily_mono:
"\<lbrakk>hereditarily P X; \<And>x. P x \<Longrightarrow> Q x\<rbrakk> \<Longrightarrow> hereditarily Q X"
by (simp add: hereditarily)
lemma hereditarily_inc:
"hereditarily P X \<Longrightarrow> P X"
by (metis hereditarily subtopology_topspace)
lemma hereditarily_subtopology:
"hereditarily P X \<Longrightarrow> hereditarily P (subtopology X S)"
by (simp add: hereditarily subtopology_subtopology)
lemma hereditarily_normal_space_continuous_closed_map_image:
assumes X: "hereditarily normal_space X" and contf: "continuous_map X Y f"
and clof: "closed_map X Y f" and fim: "f ` (topspace X) = topspace Y"
shows "hereditarily normal_space Y"
unfolding hereditarily_def
proof (intro strip)
fix T
assume "T \<subseteq> topspace Y"
then have nx: "normal_space (subtopology X {x \<in> topspace X. f x \<in> T})"
by (meson X hereditarily)
moreover have "continuous_map (subtopology X {x \<in> topspace X. f x \<in> T}) (subtopology Y T) f"
by (simp add: contf continuous_map_from_subtopology continuous_map_in_subtopology image_subset_iff)
moreover have "closed_map (subtopology X {x \<in> topspace X. f x \<in> T}) (subtopology Y T) f"
by (simp add: clof closed_map_restriction)
ultimately show "normal_space (subtopology Y T)"
using fim normal_space_continuous_closed_map_image by fastforce
qed
lemma homeomorphic_hereditarily_normal_space:
"X homeomorphic_space Y
\<Longrightarrow> (hereditarily normal_space X \<longleftrightarrow> hereditarily normal_space Y)"
by (meson hereditarily_normal_space_continuous_closed_map_image homeomorphic_eq_everything_map
homeomorphic_space homeomorphic_space_sym)
lemma hereditarily_normal_space_retraction_map_image:
"\<lbrakk>retraction_map X Y r; hereditarily normal_space X\<rbrakk> \<Longrightarrow> hereditarily normal_space Y"
by (smt (verit) hereditarily_subtopology hereditary_imp_retractive_property homeomorphic_hereditarily_normal_space)
subsection\<open>Limits in a topological space\<close>
lemma limitin_const_iff:
assumes "t1_space X" "\<not> trivial_limit F"
shows "limitin X (\<lambda>k. a) l F \<longleftrightarrow> l \<in> topspace X \<and> a = l" (is "?lhs=?rhs")
proof
assume ?lhs then show ?rhs
using assms unfolding limitin_def t1_space_def by (metis eventually_const openin_topspace)
next
assume ?rhs then show ?lhs
using assms by (auto simp: limitin_def t1_space_def)
qed
lemma compactin_sequence_with_limit:
assumes lim: "limitin X \<sigma> l sequentially" and "S \<subseteq> range \<sigma>" and SX: "S \<subseteq> topspace X"
shows "compactin X (insert l S)"
unfolding compactin_def
proof (intro conjI strip)
show "insert l S \<subseteq> topspace X"
by (meson SX insert_subset lim limitin_topspace)
fix \<U>
assume \<section>: "Ball \<U> (openin X) \<and> insert l S \<subseteq> \<Union> \<U>"
have "\<exists>V. finite V \<and> V \<subseteq> \<U> \<and> (\<exists>t \<in> V. l \<in> t) \<and> S \<subseteq> \<Union> V"
if *: "\<forall>x \<in> S. \<exists>T \<in> \<U>. x \<in> T" and "T \<in> \<U>" "l \<in> T" for T
proof -
obtain V where V: "\<And>x. x \<in> S \<Longrightarrow> V x \<in> \<U> \<and> x \<in> V x"
using * by metis
obtain N where N: "\<And>n. N \<le> n \<Longrightarrow> \<sigma> n \<in> T"
by (meson "\<section>" \<open>T \<in> \<U>\<close> \<open>l \<in> T\<close> lim limitin_sequentially)
show ?thesis
proof (intro conjI exI)
have "x \<in> T"
if "x \<in> S" and "\<forall>A. (\<forall>x \<in> S. (\<forall>n\<le>N. x \<noteq> \<sigma> n) \<or> A \<noteq> V x) \<or> x \<notin> A" for x
by (metis (no_types) N V that assms(2) imageE nle_le subsetD)
then show "S \<subseteq> \<Union> (insert T (V ` (S \<inter> \<sigma> ` {0..N})))"
by force
qed (use V that in auto)
qed
then show "\<exists>\<F>. finite \<F> \<and> \<F> \<subseteq> \<U> \<and> insert l S \<subseteq> \<Union> \<F>"
by (smt (verit, best) Union_iff \<section> insert_subset subsetD)
qed
lemma limitin_Hausdorff_unique:
assumes "limitin X f l1 F" "limitin X f l2 F" "\<not> trivial_limit F" "Hausdorff_space X"
shows "l1 = l2"
proof (rule ccontr)
assume "l1 \<noteq> l2"
with assms obtain U V where "openin X U" "openin X V" "l1 \<in> U" "l2 \<in> V" "disjnt U V"
by (metis Hausdorff_space_def limitin_topspace)
then have "eventually (\<lambda>x. f x \<in> U) F" "eventually (\<lambda>x. f x \<in> V) F"
using assms by (fastforce simp: limitin_def)+
then have "\<exists>x. f x \<in> U \<and> f x \<in> V"
using assms eventually_elim2 filter_eq_iff by fastforce
with assms \<open>disjnt U V\<close> show False
by (meson disjnt_iff)
qed
lemma limitin_kc_unique:
assumes "kc_space X" and lim1: "limitin X f l1 sequentially" and lim2: "limitin X f l2 sequentially"
shows "l1 = l2"
proof (rule ccontr)
assume "l1 \<noteq> l2"
define A where "A \<equiv> insert l1 (range f - {l2})"
have "l1 \<in> topspace X"
using lim1 limitin_def by fastforce
moreover have "compactin X (insert l1 (topspace X \<inter> (range f - {l2})))"
by (meson Diff_subset compactin_sequence_with_limit inf_le1 inf_le2 lim1 subset_trans)
ultimately have "compactin X (topspace X \<inter> A)"
by (simp add: A_def)
then have OXA: "openin X (topspace X - A)"
by (metis Diff_Diff_Int Diff_subset \<open>kc_space X\<close> kc_space_def openin_closedin_eq)
have "l2 \<in> topspace X - A"
using \<open>l1 \<noteq> l2\<close> A_def lim2 limitin_topspace by fastforce
then have "\<forall>\<^sub>F x in sequentially. f x = l2"
using limitinD [OF lim2 OXA] by (auto simp: A_def eventually_conj_iff)
then show False
using limitin_transform_eventually [OF _ lim1]
limitin_const_iff [OF kc_imp_t1_space trivial_limit_sequentially]
using \<open>l1 \<noteq> l2\<close> \<open>kc_space X\<close> by fastforce
qed
lemma limitin_closedin:
assumes lim: "limitin X f l F"
and "closedin X S" and ev: "eventually (\<lambda>x. f x \<in> S) F" "\<not> trivial_limit F"
shows "l \<in> S"
proof (rule ccontr)
assume "l \<notin> S"
have "\<forall>\<^sub>F x in F. f x \<in> topspace X - S"
by (metis Diff_iff \<open>l \<notin> S\<close> \<open>closedin X S\<close> closedin_def lim limitin_def)
with ev eventually_elim2 trivial_limit_def show False
by force
qed
subsection\<open>Quasi-components\<close>
definition quasi_component_of :: "'a topology \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"
where
"quasi_component_of X x y \<equiv>
x \<in> topspace X \<and> y \<in> topspace X \<and>
(\<forall>T. closedin X T \<and> openin X T \<longrightarrow> (x \<in> T \<longleftrightarrow> y \<in> T))"
abbreviation "quasi_component_of_set S x \<equiv> Collect (quasi_component_of S x)"
definition quasi_components_of :: "'a topology \<Rightarrow> ('a set) set"
where
"quasi_components_of X = quasi_component_of_set X ` topspace X"
lemma quasi_component_in_topspace:
"quasi_component_of X x y \<Longrightarrow> x \<in> topspace X \<and> y \<in> topspace X"
by (simp add: quasi_component_of_def)
lemma quasi_component_of_refl [simp]:
"quasi_component_of X x x \<longleftrightarrow> x \<in> topspace X"
by (simp add: quasi_component_of_def)
lemma quasi_component_of_sym:
"quasi_component_of X x y \<longleftrightarrow> quasi_component_of X y x"
by (meson quasi_component_of_def)
lemma quasi_component_of_trans:
"\<lbrakk>quasi_component_of X x y; quasi_component_of X y z\<rbrakk> \<Longrightarrow> quasi_component_of X x z"
by (simp add: quasi_component_of_def)
lemma quasi_component_of_subset_topspace:
"quasi_component_of_set X x \<subseteq> topspace X"
using quasi_component_of_def by fastforce
lemma quasi_component_of_eq_empty:
"quasi_component_of_set X x = {} \<longleftrightarrow> (x \<notin> topspace X)"
using quasi_component_of_def by fastforce
lemma quasi_component_of:
"quasi_component_of X x y \<longleftrightarrow>
x \<in> topspace X \<and> y \<in> topspace X \<and> (\<forall>T. x \<in> T \<and> closedin X T \<and> openin X T \<longrightarrow> y \<in> T)"
unfolding quasi_component_of_def by (metis Diff_iff closedin_def openin_closedin_eq)
lemma quasi_component_of_alt:
"quasi_component_of X x y \<longleftrightarrow>
x \<in> topspace X \<and> y \<in> topspace X \<and>
\<not> (\<exists>U V. openin X U \<and> openin X V \<and> U \<union> V = topspace X \<and> disjnt U V \<and> x \<in> U \<and> y \<in> V)"
(is "?lhs = ?rhs")
proof
show "?lhs \<Longrightarrow> ?rhs"
unfolding quasi_component_of_def
by (metis disjnt_iff separatedin_full separatedin_open_sets)
show "?rhs \<Longrightarrow> ?lhs"
unfolding quasi_component_of_def
by (metis Diff_disjoint Diff_iff Un_Diff_cancel closedin_def disjnt_def inf_commute sup.orderE sup_commute)
qed
lemma quasi_components_lepoll_topspace: "quasi_components_of X \<lesssim> topspace X"
by (simp add: image_lepoll quasi_components_of_def)
lemma quasi_component_of_separated:
"quasi_component_of X x y \<longleftrightarrow>
x \<in> topspace X \<and> y \<in> topspace X \<and>
\<not> (\<exists>U V. separatedin X U V \<and> U \<union> V = topspace X \<and> x \<in> U \<and> y \<in> V)"
by (meson quasi_component_of_alt separatedin_full separatedin_open_sets)
lemma quasi_component_of_subtopology:
"quasi_component_of (subtopology X s) x y \<Longrightarrow> quasi_component_of X x y"
unfolding quasi_component_of_def
by (simp add: closedin_subtopology) (metis Int_iff inf_commute openin_subtopology_Int2)
lemma quasi_component_of_mono:
"quasi_component_of (subtopology X S) x y \<and> S \<subseteq> T
\<Longrightarrow> quasi_component_of (subtopology X T) x y"
by (metis inf.absorb_iff2 quasi_component_of_subtopology subtopology_subtopology)
lemma quasi_component_of_equiv:
"quasi_component_of X x y \<longleftrightarrow>
x \<in> topspace X \<and> y \<in> topspace X \<and> quasi_component_of X x = quasi_component_of X y"
using quasi_component_of_def by fastforce
lemma quasi_component_of_disjoint [simp]:
"disjnt (quasi_component_of_set X x) (quasi_component_of_set X y) \<longleftrightarrow> \<not> (quasi_component_of X x y)"
by (metis disjnt_iff quasi_component_of_equiv mem_Collect_eq)
lemma quasi_component_of_eq:
"quasi_component_of X x = quasi_component_of X y \<longleftrightarrow>
(x \<notin> topspace X \<and> y \<notin> topspace X)
\<or> x \<in> topspace X \<and> y \<in> topspace X \<and> quasi_component_of X x y"
by (metis Collect_empty_eq_bot quasi_component_of_eq_empty quasi_component_of_equiv)
lemma topspace_imp_quasi_components_of:
assumes "x \<in> topspace X"
obtains C where "C \<in> quasi_components_of X" "x \<in> C"
by (metis assms imageI mem_Collect_eq quasi_component_of_refl quasi_components_of_def)
lemma Union_quasi_components_of: "\<Union> (quasi_components_of X) = topspace X"
by (auto simp: quasi_components_of_def quasi_component_of_def)
lemma pairwise_disjoint_quasi_components_of:
"pairwise disjnt (quasi_components_of X)"
by (auto simp: quasi_components_of_def quasi_component_of_def disjoint_def)
lemma complement_quasi_components_of_Union:
assumes "C \<in> quasi_components_of X"
shows "topspace X - C = \<Union> (quasi_components_of X - {C})" (is "?lhs = ?rhs")
proof
show "?lhs \<subseteq> ?rhs"
using Union_quasi_components_of by fastforce
show "?rhs \<subseteq> ?lhs"
using assms
using quasi_component_of_equiv by (fastforce simp add: quasi_components_of_def image_iff subset_iff)
qed
lemma nonempty_quasi_components_of:
"C \<in> quasi_components_of X \<Longrightarrow> C \<noteq> {}"
by (metis imageE quasi_component_of_eq_empty quasi_components_of_def)
lemma quasi_components_of_subset:
"C \<in> quasi_components_of X \<Longrightarrow> C \<subseteq> topspace X"
using Union_quasi_components_of by force
lemma quasi_component_in_quasi_components_of:
"quasi_component_of_set X a \<in> quasi_components_of X \<longleftrightarrow> a \<in> topspace X"
by (metis (no_types, lifting) image_iff quasi_component_of_eq_empty quasi_components_of_def)
lemma quasi_components_of_eq_empty [simp]:
"quasi_components_of X = {} \<longleftrightarrow> X = trivial_topology"
by (simp add: quasi_components_of_def)
lemma quasi_components_of_empty_space [simp]:
"quasi_components_of trivial_topology = {}"
by simp
lemma quasi_component_of_set:
"quasi_component_of_set X x =
(if x \<in> topspace X
then \<Inter> {t. closedin X t \<and> openin X t \<and> x \<in> t}
else {})"
by (auto simp: quasi_component_of)
lemma closedin_quasi_component_of: "closedin X (quasi_component_of_set X x)"
by (auto simp: quasi_component_of_set)
lemma closedin_quasi_components_of:
"C \<in> quasi_components_of X \<Longrightarrow> closedin X C"
by (auto simp: quasi_components_of_def closedin_quasi_component_of)
lemma openin_finite_quasi_components:
"\<lbrakk>finite(quasi_components_of X); C \<in> quasi_components_of X\<rbrakk> \<Longrightarrow> openin X C"
apply (simp add:openin_closedin_eq quasi_components_of_subset complement_quasi_components_of_Union)
by (meson DiffD1 closedin_Union closedin_quasi_components_of finite_Diff)
lemma quasi_component_of_eq_overlap:
"quasi_component_of X x = quasi_component_of X y \<longleftrightarrow>
(x \<notin> topspace X \<and> y \<notin> topspace X) \<or>
\<not> (quasi_component_of_set X x \<inter> quasi_component_of_set X y = {})"
using quasi_component_of_equiv by fastforce
lemma quasi_component_of_nonoverlap:
"quasi_component_of_set X x \<inter> quasi_component_of_set X y = {} \<longleftrightarrow>
(x \<notin> topspace X) \<or> (y \<notin> topspace X) \<or>
\<not> (quasi_component_of X x = quasi_component_of X y)"
by (metis inf.idem quasi_component_of_eq_empty quasi_component_of_eq_overlap)
lemma quasi_component_of_overlap:
"\<not> (quasi_component_of_set X x \<inter> quasi_component_of_set X y = {}) \<longleftrightarrow>
x \<in> topspace X \<and> y \<in> topspace X \<and> quasi_component_of X x = quasi_component_of X y"
by (meson quasi_component_of_nonoverlap)
lemma quasi_components_of_disjoint:
"\<lbrakk>C \<in> quasi_components_of X; D \<in> quasi_components_of X\<rbrakk> \<Longrightarrow> disjnt C D \<longleftrightarrow> C \<noteq> D"
by (metis disjnt_self_iff_empty nonempty_quasi_components_of pairwiseD pairwise_disjoint_quasi_components_of)
lemma quasi_components_of_overlap:
"\<lbrakk>C \<in> quasi_components_of X; D \<in> quasi_components_of X\<rbrakk> \<Longrightarrow> \<not> (C \<inter> D = {}) \<longleftrightarrow> C = D"
by (metis disjnt_def quasi_components_of_disjoint)
lemma pairwise_separated_quasi_components_of:
"pairwise (separatedin X) (quasi_components_of X)"
by (metis closedin_quasi_components_of pairwise_def pairwise_disjoint_quasi_components_of separatedin_closed_sets)
lemma finite_quasi_components_of_finite:
"finite(topspace X) \<Longrightarrow> finite(quasi_components_of X)"
by (simp add: Union_quasi_components_of finite_UnionD)
lemma connected_imp_quasi_component_of:
assumes "connected_component_of X x y"
shows "quasi_component_of X x y"
proof -
have "x \<in> topspace X" "y \<in> topspace X"
by (meson assms connected_component_of_equiv)+
with assms show ?thesis
apply (clarsimp simp add: quasi_component_of connected_component_of_def)
by (meson connectedin_clopen_cases disjnt_iff subsetD)
qed
lemma connected_component_subset_quasi_component_of:
"connected_component_of_set X x \<subseteq> quasi_component_of_set X x"
using connected_imp_quasi_component_of by force
lemma quasi_component_as_connected_component_Union:
"quasi_component_of_set X x =
\<Union> (connected_component_of_set X ` quasi_component_of_set X x)"
(is "?lhs = ?rhs")
proof
show "?lhs \<subseteq> ?rhs"
using connected_component_of_refl quasi_component_of by fastforce
show "?rhs \<subseteq> ?lhs"
apply (rule SUP_least)
by (simp add: connected_component_subset_quasi_component_of quasi_component_of_equiv)
qed
lemma quasi_components_as_connected_components_Union:
assumes "C \<in> quasi_components_of X"
obtains \<T> where "\<T> \<subseteq> connected_components_of X" "\<Union>\<T> = C"
proof -
obtain x where "x \<in> topspace X" and Ceq: "C = quasi_component_of_set X x"
by (metis assms imageE quasi_components_of_def)
define \<T> where "\<T> \<equiv> connected_component_of_set X ` quasi_component_of_set X x"
show thesis
proof
show "\<T> \<subseteq> connected_components_of X"
by (simp add: \<T>_def connected_components_of_def image_mono quasi_component_of_subset_topspace)
show "\<Union>\<T> = C"
by (metis \<T>_def Ceq quasi_component_as_connected_component_Union)
qed
qed
lemma path_imp_quasi_component_of:
"path_component_of X x y \<Longrightarrow> quasi_component_of X x y"
by (simp add: connected_imp_quasi_component_of path_imp_connected_component_of)
lemma path_component_subset_quasi_component_of:
"path_component_of_set X x \<subseteq> quasi_component_of_set X x"
by (simp add: Collect_mono path_imp_quasi_component_of)
lemma connected_space_iff_quasi_component:
"connected_space X \<longleftrightarrow> (\<forall>x \<in> topspace X. \<forall>y \<in> topspace X. quasi_component_of X x y)"
unfolding connected_space_clopen_in closedin_def quasi_component_of
by blast
lemma connected_space_imp_quasi_component_of:
" \<lbrakk>connected_space X; a \<in> topspace X; b \<in> topspace X\<rbrakk> \<Longrightarrow> quasi_component_of X a b"
by (simp add: connected_space_iff_quasi_component)
lemma connected_space_quasi_component_set:
"connected_space X \<longleftrightarrow> (\<forall>x \<in> topspace X. quasi_component_of_set X x = topspace X)"
by (metis Ball_Collect connected_space_iff_quasi_component quasi_component_of_subset_topspace subset_antisym)
lemma connected_space_iff_quasi_components_eq:
"connected_space X \<longleftrightarrow>
(\<forall>C \<in> quasi_components_of X. \<forall>D \<in> quasi_components_of X. C = D)"
apply (simp add: quasi_components_of_def)
by (metis connected_space_iff_quasi_component mem_Collect_eq quasi_component_of_equiv)
lemma quasi_components_of_subset_sing:
"quasi_components_of X \<subseteq> {S} \<longleftrightarrow> connected_space X \<and> (X = trivial_topology \<or> topspace X = S)"
proof (cases "quasi_components_of X = {}")
case True
then show ?thesis
by (simp add: subset_singleton_iff)
next
case False
then show ?thesis
apply (simp add: connected_space_iff_quasi_components_eq subset_iff Ball_def)
by (metis False Union_quasi_components_of ccpo_Sup_singleton insert_iff is_singletonE is_singletonI')
qed
lemma connected_space_iff_quasi_components_subset_sing:
"connected_space X \<longleftrightarrow> (\<exists>a. quasi_components_of X \<subseteq> {a})"
by (simp add: quasi_components_of_subset_sing)
lemma quasi_components_of_eq_singleton:
"quasi_components_of X = {S} \<longleftrightarrow>
connected_space X \<and> \<not> (X = trivial_topology) \<and> S = topspace X"
by (metis empty_not_insert quasi_components_of_eq_empty quasi_components_of_subset_sing subset_singleton_iff)
lemma quasi_components_of_connected_space:
"connected_space X
\<Longrightarrow> quasi_components_of X = (if X = trivial_topology then {} else {topspace X})"
by (simp add: quasi_components_of_eq_singleton)
lemma separated_between_singletons:
"separated_between X {x} {y} \<longleftrightarrow>
x \<in> topspace X \<and> y \<in> topspace X \<and> \<not> (quasi_component_of X x y)"
proof (cases "x \<in> topspace X \<and> y \<in> topspace X")
case True
then show ?thesis
by (auto simp add: separated_between_def quasi_component_of_alt)
qed (use separated_between_imp_subset in blast)
lemma quasi_component_nonseparated:
"quasi_component_of X x y \<longleftrightarrow> x \<in> topspace X \<and> y \<in> topspace X \<and> \<not> (separated_between X {x} {y})"
by (metis quasi_component_of_equiv separated_between_singletons)
lemma separated_between_quasi_component_pointwise_left:
assumes "C \<in> quasi_components_of X"
shows "separated_between X C S \<longleftrightarrow> (\<exists>x \<in> C. separated_between X {x} S)" (is "?lhs = ?rhs")
proof
show "?lhs \<Longrightarrow> ?rhs"
using assms quasi_components_of_disjoint separated_between_mono by fastforce
next
assume ?rhs
then obtain y where "separated_between X {y} S" and "y \<in> C"
by metis
with assms show ?lhs
by (force simp add: separated_between quasi_components_of_def quasi_component_of_def)
qed
lemma separated_between_quasi_component_pointwise_right:
"C \<in> quasi_components_of X \<Longrightarrow> separated_between X S C \<longleftrightarrow> (\<exists>x \<in> C. separated_between X S {x})"
by (simp add: separated_between_quasi_component_pointwise_left separated_between_sym)
lemma separated_between_quasi_component_point:
assumes "C \<in> quasi_components_of X"
shows "separated_between X C {x} \<longleftrightarrow> x \<in> topspace X - C" (is "?lhs = ?rhs")
proof
show "?lhs \<Longrightarrow> ?rhs"
by (meson DiffI disjnt_insert2 insert_subset separated_between_imp_disjoint separated_between_imp_subset)
next
assume ?rhs
with assms show ?lhs
unfolding quasi_components_of_def image_iff Diff_iff separated_between_quasi_component_pointwise_left [OF assms]
by (metis mem_Collect_eq quasi_component_of_refl separated_between_singletons)
qed
lemma separated_between_point_quasi_component:
"C \<in> quasi_components_of X \<Longrightarrow> separated_between X {x} C \<longleftrightarrow> x \<in> topspace X - C"
by (simp add: separated_between_quasi_component_point separated_between_sym)
lemma separated_between_quasi_component_compact:
"\<lbrakk>C \<in> quasi_components_of X; compactin X K\<rbrakk> \<Longrightarrow> (separated_between X C K \<longleftrightarrow> disjnt C K)"
unfolding disjnt_iff
using compactin_subset_topspace quasi_components_of_subset separated_between_pointwise_right separated_between_quasi_component_point by fastforce
lemma separated_between_compact_quasi_component:
"\<lbrakk>compactin X K; C \<in> quasi_components_of X\<rbrakk> \<Longrightarrow> separated_between X K C \<longleftrightarrow> disjnt K C"
using disjnt_sym separated_between_quasi_component_compact separated_between_sym by blast
lemma separated_between_quasi_components:
assumes C: "C \<in> quasi_components_of X" and D: "D \<in> quasi_components_of X"
shows "separated_between X C D \<longleftrightarrow> disjnt C D" (is "?lhs = ?rhs")
proof
show "?lhs \<Longrightarrow> ?rhs"
by (simp add: separated_between_imp_disjoint)
next
assume ?rhs
obtain x y where x: "C = quasi_component_of_set X x" and "x \<in> C"
and y: "D = quasi_component_of_set X y" and "y \<in> D"
using assms by (auto simp: quasi_components_of_def)
then have "separated_between X {x} {y}"
using \<open>disjnt C D\<close> separated_between_singletons by fastforce
with \<open>x \<in> C\<close> \<open>y \<in> D\<close> show ?lhs
by (auto simp: assms separated_between_quasi_component_pointwise_left separated_between_quasi_component_pointwise_right)
qed
lemma quasi_eq_connected_component_of_eq:
"quasi_component_of X x = connected_component_of X x \<longleftrightarrow>
connectedin X (quasi_component_of_set X x)" (is "?lhs = ?rhs")
proof (cases "x \<in> topspace X")
case True
show ?thesis
proof
show "?lhs \<Longrightarrow> ?rhs"
by (simp add: connectedin_connected_component_of)
next
assume ?rhs
then have "\<And>y. quasi_component_of X x y = connected_component_of X x y"
by (metis connected_component_of_def connected_imp_quasi_component_of mem_Collect_eq quasi_component_of_equiv)
then show ?lhs
by force
qed
next
case False
then show ?thesis
by (metis Collect_empty_eq_bot connected_component_of_eq_empty connectedin_empty quasi_component_of_eq_empty)
qed
lemma connected_quasi_component_of:
assumes "C \<in> quasi_components_of X"
shows "C \<in> connected_components_of X \<longleftrightarrow> connectedin X C" (is "?lhs = ?rhs")
proof
show "?lhs \<Longrightarrow> ?rhs"
using assms
by (simp add: connectedin_connected_components_of)
next
assume ?rhs
with assms show ?lhs
unfolding quasi_components_of_def connected_components_of_def image_iff
by (metis quasi_eq_connected_component_of_eq)
qed
lemma quasi_component_of_clopen_cases:
"\<lbrakk>C \<in> quasi_components_of X; closedin X T; openin X T\<rbrakk> \<Longrightarrow> C \<subseteq> T \<or> disjnt C T"
by (smt (verit) disjnt_iff image_iff mem_Collect_eq quasi_component_of_def quasi_components_of_def subset_iff)
lemma quasi_components_of_set:
assumes "C \<in> quasi_components_of X"
shows "\<Inter> {T. closedin X T \<and> openin X T \<and> C \<subseteq> T} = C" (is "?lhs = ?rhs")
proof
have "x \<in> C" if "x \<in> \<Inter> {T. closedin X T \<and> openin X T \<and> C \<subseteq> T}" for x
proof (rule ccontr)
assume "x \<notin> C"
have "x \<in> topspace X"
using assms quasi_components_of_subset that by force
then have "separated_between X C {x}"
by (simp add: \<open>x \<notin> C\<close> assms separated_between_quasi_component_point)
with that show False
by (auto simp: separated_between)
qed
then show "?lhs \<subseteq> ?rhs"
by auto
qed blast
lemma open_quasi_eq_connected_components_of:
assumes "openin X C"
shows "C \<in> quasi_components_of X \<longleftrightarrow> C \<in> connected_components_of X" (is "?lhs = ?rhs")
proof (cases "closedin X C")
case True
show ?thesis
proof
assume L: ?lhs
have "T = {} \<or> T = topspace X \<inter> C"
if "openin (subtopology X C) T" "closedin (subtopology X C) T" for T
proof -
have "C \<subseteq> T \<or> disjnt C T"
by (meson L True assms closedin_trans_full openin_trans_full quasi_component_of_clopen_cases that)
with that show ?thesis
by (metis Int_absorb2 True closedin_imp_subset closure_of_subset_eq disjnt_def inf_absorb2)
qed
with L assms show "?rhs"
by (simp add: connected_quasi_component_of connected_space_clopen_in connectedin_def openin_subset)
next
assume ?rhs
then obtain x where "x \<in> topspace X" and x: "C = connected_component_of_set X x"
by (metis connected_components_of_def imageE)
have "C = quasi_component_of_set X x"
using True assms connected_component_of_refl connected_imp_quasi_component_of quasi_component_of_def x by fastforce
then show ?lhs
using \<open>x \<in> topspace X\<close> quasi_components_of_def by fastforce
qed
next
case False
then show ?thesis
using closedin_connected_components_of closedin_quasi_components_of by blast
qed
lemma quasi_component_of_continuous_image:
assumes f: "continuous_map X Y f" and qc: "quasi_component_of X x y"
shows "quasi_component_of Y (f x) (f y)"
unfolding quasi_component_of_def
proof (intro strip conjI)
show "f x \<in> topspace Y" "f y \<in> topspace Y"
using assms by (simp_all add: continuous_map_def quasi_component_of_def Pi_iff)
fix T
assume "closedin Y T \<and> openin Y T"
with assms show "(f x \<in> T) = (f y \<in> T)"
by (smt (verit) continuous_map_closedin continuous_map_def mem_Collect_eq quasi_component_of_def)
qed
lemma quasi_component_of_discrete_topology:
"quasi_component_of_set (discrete_topology U) x = (if x \<in> U then {x} else {})"
proof -
have "quasi_component_of_set (discrete_topology U) y = {y}" if "y \<in> U" for y
using that
apply (simp add: set_eq_iff quasi_component_of_def)
by (metis Set.set_insert insertE subset_insertI)
then show ?thesis
by (simp add: quasi_component_of)
qed
lemma quasi_components_of_discrete_topology:
"quasi_components_of (discrete_topology U) = (\<lambda>x. {x}) ` U"
by (auto simp add: quasi_components_of_def quasi_component_of_discrete_topology)
lemma homeomorphic_map_quasi_component_of:
assumes hmf: "homeomorphic_map X Y f" and "x \<in> topspace X"
shows "quasi_component_of_set Y (f x) = f ` (quasi_component_of_set X x)"
proof -
obtain g where hmg: "homeomorphic_map Y X g"
and contf: "continuous_map X Y f" and contg: "continuous_map Y X g"
and fg: "(\<forall>x \<in> topspace X. g(f x) = x) \<and> (\<forall>y \<in> topspace Y. f(g y) = y)"
by (smt (verit, best) hmf homeomorphic_map_maps homeomorphic_maps_def)
show ?thesis
proof
show "quasi_component_of_set Y (f x) \<subseteq> f ` quasi_component_of_set X x"
using quasi_component_of_continuous_image [OF contg]
\<open>x \<in> topspace X\<close> fg image_iff quasi_component_of_subset_topspace by fastforce
show "f ` quasi_component_of_set X x \<subseteq> quasi_component_of_set Y (f x)"
using quasi_component_of_continuous_image [OF contf] by blast
qed
qed
lemma homeomorphic_map_quasi_components_of:
assumes "homeomorphic_map X Y f"
shows "quasi_components_of Y = image (image f) (quasi_components_of X)"
using assms
proof -
have "\<exists>x\<in>topspace X. quasi_component_of_set Y y = f ` quasi_component_of_set X x"
if "y \<in> topspace Y" for y
by (metis that assms homeomorphic_imp_surjective_map homeomorphic_map_quasi_component_of image_iff)
moreover have "\<exists>x\<in>topspace Y. f ` quasi_component_of_set X u = quasi_component_of_set Y x"
if "u \<in> topspace X" for u
by (metis that assms homeomorphic_imp_surjective_map homeomorphic_map_quasi_component_of imageI)
ultimately show ?thesis
by (auto simp: quasi_components_of_def image_iff)
qed
lemma openin_quasi_component_of_locally_connected_space:
assumes "locally_connected_space X"
shows "openin X (quasi_component_of_set X x)"
proof -
have *: "openin X (connected_component_of_set X x)"
by (simp add: assms openin_connected_component_of_locally_connected_space)
moreover have "connected_component_of_set X x = quasi_component_of_set X x"
using * closedin_connected_component_of connected_component_of_refl connected_imp_quasi_component_of
quasi_component_of_def by fastforce
ultimately show ?thesis
by simp
qed
lemma openin_quasi_components_of_locally_connected_space:
"locally_connected_space X \<and> c \<in> quasi_components_of X
\<Longrightarrow> openin X c"
by (smt (verit, best) image_iff openin_quasi_component_of_locally_connected_space quasi_components_of_def)
lemma quasi_eq_connected_components_of_alt:
"quasi_components_of X = connected_components_of X \<longleftrightarrow> (\<forall>C \<in> quasi_components_of X. connectedin X C)"
(is "?lhs = ?rhs")
proof
assume R: ?rhs
moreover have "connected_components_of X \<subseteq> quasi_components_of X"
using R unfolding quasi_components_of_def connected_components_of_def
by (force simp flip: quasi_eq_connected_component_of_eq)
ultimately show ?lhs
using connected_quasi_component_of by blast
qed (use connected_quasi_component_of in blast)
lemma connected_subset_quasi_components_of_pointwise:
"connected_components_of X \<subseteq> quasi_components_of X \<longleftrightarrow>
(\<forall>x \<in> topspace X. quasi_component_of X x = connected_component_of X x)"
(is "?lhs = ?rhs")
proof
assume L: ?lhs
have "connectedin X (quasi_component_of_set X x)" if "x \<in> topspace X" for x
proof -
have "\<exists>y\<in>topspace X. connected_component_of_set X x = quasi_component_of_set X y"
using L that by (force simp: quasi_components_of_def connected_components_of_def image_subset_iff)
then show ?thesis
by (metis connected_component_of_equiv connectedin_connected_component_of mem_Collect_eq quasi_component_of_eq)
qed
then show ?rhs
by (simp add: quasi_eq_connected_component_of_eq)
qed (simp add: connected_components_of_def quasi_components_of_def)
lemma quasi_subset_connected_components_of_pointwise:
"quasi_components_of X \<subseteq> connected_components_of X \<longleftrightarrow>
(\<forall>x \<in> topspace X. quasi_component_of X x = connected_component_of X x)"
by (simp add: connected_quasi_component_of image_subset_iff quasi_components_of_def quasi_eq_connected_component_of_eq)
lemma quasi_eq_connected_components_of_pointwise:
"quasi_components_of X = connected_components_of X \<longleftrightarrow>
(\<forall>x \<in> topspace X. quasi_component_of X x = connected_component_of X x)"
using connected_subset_quasi_components_of_pointwise quasi_subset_connected_components_of_pointwise by fastforce
lemma quasi_eq_connected_components_of_pointwise_alt:
"quasi_components_of X = connected_components_of X \<longleftrightarrow>
(\<forall>x. quasi_component_of X x = connected_component_of X x)"
unfolding quasi_eq_connected_components_of_pointwise
by (metis connectedin_empty quasi_component_of_eq_empty quasi_eq_connected_component_of_eq)
lemma quasi_eq_connected_components_of_inclusion:
"quasi_components_of X = connected_components_of X \<longleftrightarrow>
connected_components_of X \<subseteq> quasi_components_of X \<or>
quasi_components_of X \<subseteq> connected_components_of X"
by (simp add: connected_subset_quasi_components_of_pointwise dual_order.eq_iff quasi_subset_connected_components_of_pointwise)
lemma quasi_eq_connected_components_of:
"finite(connected_components_of X) \<or>
finite(quasi_components_of X) \<or>
locally_connected_space X \<or>
compact_space X \<and> (Hausdorff_space X \<or> regular_space X \<or> normal_space X)
\<Longrightarrow> quasi_components_of X = connected_components_of X"
proof (elim disjE)
show "quasi_components_of X = connected_components_of X"
if "finite (connected_components_of X)"
unfolding quasi_eq_connected_components_of_inclusion
using that open_in_finite_connected_components open_quasi_eq_connected_components_of by blast
show "quasi_components_of X = connected_components_of X"
if "finite (quasi_components_of X)"
unfolding quasi_eq_connected_components_of_inclusion
using that open_quasi_eq_connected_components_of openin_finite_quasi_components by blast
show "quasi_components_of X = connected_components_of X"
if "locally_connected_space X"
unfolding quasi_eq_connected_components_of_inclusion
using that open_quasi_eq_connected_components_of openin_quasi_components_of_locally_connected_space by auto
show "quasi_components_of X = connected_components_of X"
if "compact_space X \<and> (Hausdorff_space X \<or> regular_space X \<or> normal_space X)"
proof -
show ?thesis
unfolding quasi_eq_connected_components_of_alt
proof (intro strip)
fix C
assume C: "C \<in> quasi_components_of X"
then have cloC: "closedin X C"
by (simp add: closedin_quasi_components_of)
have "normal_space X"
using that compact_Hausdorff_or_regular_imp_normal_space by blast
show "connectedin X C"
proof (clarsimp simp add: connectedin_def connected_space_closedin_eq closedin_closed_subtopology cloC closedin_subset [OF cloC])
fix S T
assume "S \<subseteq> C" and "closedin X S" and "S \<inter> T = {}" and SUT: "S \<union> T = topspace X \<inter> C"
and T: "T \<subseteq> C" "T \<noteq> {}" and "closedin X T"
with \<open>normal_space X\<close> obtain U V where UV: "openin X U" "openin X V" "S \<subseteq> U" "T \<subseteq> V" "disjnt U V"
by (meson disjnt_def normal_space_def)
moreover have "compactin X (topspace X - (U \<union> V))"
using UV that by (intro closedin_compact_space closedin_diff openin_Un) auto
ultimately have "separated_between X C (topspace X - (U \<union> V)) \<longleftrightarrow> disjnt C (topspace X - (U \<union> V))"
by (simp add: \<open>C \<in> quasi_components_of X\<close> separated_between_quasi_component_compact)
moreover have "disjnt C (topspace X - (U \<union> V))"
using UV SUT disjnt_def by fastforce
ultimately have "separated_between X C (topspace X - (U \<union> V))"
by simp
then obtain A B where "openin X A" "openin X B" "A \<union> B = topspace X" "disjnt A B" "C \<subseteq> A"
and subB: "topspace X - (U \<union> V) \<subseteq> B"
by (meson separated_between_def)
have "B \<union> U = topspace X - (A \<inter> V)"
proof
show "B \<union> U \<subseteq> topspace X - A \<inter> V"
using \<open>openin X U\<close> \<open>disjnt U V\<close> \<open>disjnt A B\<close> \<open>openin X B\<close> disjnt_iff openin_closedin_eq by fastforce
show "topspace X - A \<inter> V \<subseteq> B \<union> U"
using \<open>A \<union> B = topspace X\<close> subB by fastforce
qed
then have "closedin X (B \<union> U)"
using \<open>openin X V\<close> \<open>openin X A\<close> by auto
then have "C \<subseteq> B \<union> U \<or> disjnt C (B \<union> U)"
using quasi_component_of_clopen_cases [OF C] \<open>openin X U\<close> \<open>openin X B\<close> by blast
with UV show "S = {}"
by (metis UnE \<open>C \<subseteq> A\<close> \<open>S \<subseteq> C\<close> T \<open>disjnt A B\<close> all_not_in_conv disjnt_Un2 disjnt_iff subset_eq)
qed
qed
qed
qed
lemma quasi_eq_connected_component_of:
"finite(connected_components_of X) \<or>
finite(quasi_components_of X) \<or>
locally_connected_space X \<or>
compact_space X \<and> (Hausdorff_space X \<or> regular_space X \<or> normal_space X)
\<Longrightarrow> quasi_component_of X x = connected_component_of X x"
by (metis quasi_eq_connected_components_of quasi_eq_connected_components_of_pointwise_alt)
subsection\<open>Additional quasicomponent and continuum properties like Boundary Bumping\<close>
lemma cut_wire_fence_theorem_gen:
assumes "compact_space X" and X: "Hausdorff_space X \<or> regular_space X \<or> normal_space X"
and S: "compactin X S" and T: "closedin X T"
and dis: "\<And>C. connectedin X C \<Longrightarrow> disjnt C S \<or> disjnt C T"
shows "separated_between X S T"
proof -
have "x \<in> topspace X" if "x \<in> S" and "T = {}" for x
using that S compactin_subset_topspace by auto
moreover have "separated_between X {x} {y}" if "x \<in> S" and "y \<in> T" for x y
proof (cases "x \<in> topspace X \<and> y \<in> topspace X")
case True
then have "\<not> connected_component_of X x y"
by (meson dis connected_component_of_def disjnt_iff that)
with True X \<open>compact_space X\<close> show ?thesis
by (metis quasi_component_nonseparated quasi_eq_connected_component_of)
next
case False
then show ?thesis
using S T compactin_subset_topspace closedin_subset that by blast
qed
ultimately show ?thesis
using assms
by (simp add: separated_between_pointwise_left separated_between_pointwise_right
closedin_compact_space closedin_subset)
qed
lemma cut_wire_fence_theorem:
"\<lbrakk>compact_space X; Hausdorff_space X; closedin X S; closedin X T;
\<And>C. connectedin X C \<Longrightarrow> disjnt C S \<or> disjnt C T\<rbrakk>
\<Longrightarrow> separated_between X S T"
by (simp add: closedin_compact_space cut_wire_fence_theorem_gen)
lemma separated_between_from_closed_subtopology:
assumes XC: "separated_between (subtopology X C) S (X frontier_of C)"
and ST: "separated_between (subtopology X C) S T"
shows "separated_between X S T"
proof -
obtain U where clo: "closedin (subtopology X C) U" and ope: "openin (subtopology X C) U"
and "S \<subseteq> U" and sub: "X frontier_of C \<union> T \<subseteq> topspace (subtopology X C) - U"
by (meson assms separated_between separated_between_Un)
then have "X frontier_of C \<union> T \<subseteq> topspace X \<inter> C - U"
by auto
have "closedin X (topspace X \<inter> C)"
by (metis XC frontier_of_restrict frontier_of_subset_eq inf_le1 separated_between_imp_subset topspace_subtopology)
then have "closedin X U"
by (metis clo closedin_closed_subtopology subtopology_restrict)
moreover have "openin (subtopology X C) U \<longleftrightarrow> openin X U \<and> U \<subseteq> C"
using disjnt_iff sub by (force intro!: openin_subset_topspace_eq)
with ope have "openin X U"
by blast
moreover have "T \<subseteq> topspace X - U"
using ope openin_closedin_eq sub by auto
ultimately show ?thesis
using \<open>S \<subseteq> U\<close> separated_between by blast
qed
lemma separated_between_from_closed_subtopology_frontier:
"separated_between (subtopology X T) S (X frontier_of T)
\<Longrightarrow> separated_between X S (X frontier_of T)"
using separated_between_from_closed_subtopology by blast
lemma separated_between_from_frontier_of_closed_subtopology:
assumes "separated_between (subtopology X T) S (X frontier_of T)"
shows "separated_between X S (topspace X - T)"
proof -
have "disjnt S (topspace X - T)"
using assms disjnt_iff separated_between_imp_subset by fastforce
then show ?thesis
by (metis Diff_subset assms frontier_of_complement separated_between_from_closed_subtopology separated_between_frontier_of_eq')
qed
lemma separated_between_compact_connected_component:
assumes "locally_compact_space X" "Hausdorff_space X"
and C: "C \<in> connected_components_of X"
and "compactin X C" "closedin X T" "disjnt C T"
shows "separated_between X C T"
proof -
have Csub: "C \<subseteq> topspace X"
by (simp add: assms(4) compactin_subset_topspace)
have "Hausdorff_space (subtopology X (topspace X - T))"
using Hausdorff_space_subtopology assms(2) by blast
moreover have "compactin (subtopology X (topspace X - T)) C"
using assms Csub by (metis Diff_Int_distrib Diff_empty compact_imp_compactin_subtopology disjnt_def le_iff_inf)
moreover have "locally_compact_space (subtopology X (topspace X - T))"
by (meson assms closedin_def locally_compact_Hausdorff_imp_regular_space locally_compact_space_open_subset)
ultimately
obtain N L where "openin X N" "compactin X L" "closedin X L" "C \<subseteq> N" "N \<subseteq> L"
and Lsub: "L \<subseteq> topspace X - T"
using \<open>Hausdorff_space X\<close> \<open>closedin X T\<close>
apply (simp add: locally_compact_space_compact_closed_compact compactin_subtopology)
by (meson closedin_def compactin_imp_closedin openin_trans_full)
then have disC: "disjnt C (topspace X - L)"
by (meson DiffD2 disjnt_iff subset_iff)
have "separated_between (subtopology X L) C (X frontier_of L)"
proof (rule cut_wire_fence_theorem)
show "compact_space (subtopology X L)"
by (simp add: \<open>compactin X L\<close> compact_space_subtopology)
show "Hausdorff_space (subtopology X L)"
by (simp add: Hausdorff_space_subtopology \<open>Hausdorff_space X\<close>)
show "closedin (subtopology X L) C"
by (meson \<open>C \<subseteq> N\<close> \<open>N \<subseteq> L\<close> \<open>Hausdorff_space X\<close> \<open>compactin X C\<close> closedin_subset_topspace compactin_imp_closedin subset_trans)
show "closedin (subtopology X L) (X frontier_of L)"
by (simp add: \<open>closedin X L\<close> closedin_frontier_of closedin_subset_topspace frontier_of_subset_closedin)
show "disjnt D C \<or> disjnt D (X frontier_of L)"
if "connectedin (subtopology X L) D" for D
proof (rule ccontr)
assume "\<not> (disjnt D C \<or> disjnt D (X frontier_of L))"
moreover have "connectedin X D"
using connectedin_subtopology that by blast
ultimately show False
using that connected_components_of_maximal [of C X D] C
apply (simp add: disjnt_iff)
by (metis Diff_eq_empty_iff \<open>C \<subseteq> N\<close> \<open>N \<subseteq> L\<close> \<open>openin X N\<close> disjoint_iff frontier_of_openin_straddle_Int(2) subsetD)
qed
qed
then have "separated_between X (X frontier_of C) (topspace X - L)"
using separated_between_from_frontier_of_closed_subtopology separated_between_frontier_of_eq by blast
with \<open>closedin X T\<close>
separated_between_frontier_of [OF Csub disC]
show ?thesis
unfolding separated_between by (smt (verit) Diff_iff Lsub closedin_subset subset_iff)
qed
lemma wilder_locally_compact_component_thm:
assumes "locally_compact_space X" "Hausdorff_space X"
and "C \<in> connected_components_of X" "compactin X C" "openin X W" "C \<subseteq> W"
obtains U V where "openin X U" "openin X V" "disjnt U V" "U \<union> V = topspace X" "C \<subseteq> U" "U \<subseteq> W"
proof -
have "closedin X (topspace X - W)"
using \<open>openin X W\<close> by blast
moreover have "disjnt C (topspace X - W)"
using \<open>C \<subseteq> W\<close> disjnt_def by fastforce
ultimately have "separated_between X C (topspace X - W)"
using separated_between_compact_connected_component assms by blast
then show thesis
by (smt (verit, del_insts) DiffI disjnt_iff openin_subset separated_between_def subset_iff that)
qed
lemma compact_quasi_eq_connected_components_of:
assumes "locally_compact_space X" "Hausdorff_space X" "compactin X C"
shows "C \<in> quasi_components_of X \<longleftrightarrow> C \<in> connected_components_of X"
proof -
have "compactin X (connected_component_of_set X x)"
if "x \<in> topspace X" "compactin X (quasi_component_of_set X x)" for x
proof (rule closed_compactin)
show "compactin X (quasi_component_of_set X x)"
by (simp add: that)
show "connected_component_of_set X x \<subseteq> quasi_component_of_set X x"
by (simp add: connected_component_subset_quasi_component_of)
show "closedin X (connected_component_of_set X x)"
by (simp add: closedin_connected_component_of)
qed
moreover have "connected_component_of X x = quasi_component_of X x"
if \<section>: "x \<in> topspace X" "compactin X (connected_component_of_set X x)" for x
proof -
have "\<And>y. connected_component_of X x y \<Longrightarrow> quasi_component_of X x y"
by (simp add: connected_imp_quasi_component_of)
moreover have False if non: "\<not> connected_component_of X x y" and quasi: "quasi_component_of X x y" for y
proof -
have "y \<in> topspace X"
by (meson quasi_component_of_equiv that)
then have "closedin X {y}"
by (simp add: \<open>Hausdorff_space X\<close> compactin_imp_closedin)
moreover have "disjnt (connected_component_of_set X x) {y}"
by (simp add: non)
moreover have "\<not> separated_between X (connected_component_of_set X x) {y}"
using \<section> quasi separated_between_pointwise_left
by (fastforce simp: quasi_component_nonseparated connected_component_of_refl)
ultimately show False
using assms by (metis \<section> connected_component_in_connected_components_of separated_between_compact_connected_component)
qed
ultimately show ?thesis
by blast
qed
ultimately show ?thesis
using \<open>compactin X C\<close> unfolding connected_components_of_def image_iff quasi_components_of_def by metis
qed
lemma boundary_bumping_theorem_closed_gen:
assumes "connected_space X" "locally_compact_space X" "Hausdorff_space X" "closedin X S"
"S \<noteq> topspace X" and C: "compactin X C" "C \<in> connected_components_of (subtopology X S)"
shows "C \<inter> X frontier_of S \<noteq> {}"
proof
assume \<section>: "C \<inter> X frontier_of S = {}"
consider "C \<noteq> {}" "X frontier_of S \<subseteq> topspace X" | "C \<subseteq> topspace X" "S = {}"
using C by (metis frontier_of_subset_topspace nonempty_connected_components_of)
then show False
proof cases
case 1
have "separated_between (subtopology X S) C (X frontier_of S)"
proof (rule separated_between_compact_connected_component)
show "compactin (subtopology X S) C"
using C compact_imp_compactin_subtopology connected_components_of_subset by fastforce
show "closedin (subtopology X S) (X frontier_of S)"
by (simp add: \<open>closedin X S\<close> closedin_frontier_of closedin_subset_topspace frontier_of_subset_closedin)
show "disjnt C (X frontier_of S)"
using \<section> by (simp add: disjnt_def)
qed (use assms Hausdorff_space_subtopology locally_compact_space_closed_subset in auto)
then have "separated_between X C (X frontier_of S)"
using separated_between_from_closed_subtopology by auto
then have "X frontier_of S = {}"
using \<open>C \<noteq> {}\<close> \<open>connected_space X\<close> connected_space_separated_between by blast
moreover have "C \<subseteq> S"
using C connected_components_of_subset by fastforce
ultimately show False
using 1 assms by (metis closedin_subset connected_space_eq_frontier_eq_empty subset_empty)
next
case 2
then show False
using C connected_components_of_eq_empty by fastforce
qed
qed
lemma boundary_bumping_theorem_closed:
assumes "connected_space X" "compact_space X" "Hausdorff_space X" "closedin X S"
"S \<noteq> topspace X" "C \<in> connected_components_of(subtopology X S)"
shows "C \<inter> X frontier_of S \<noteq> {}"
by (meson assms boundary_bumping_theorem_closed_gen closedin_compact_space closedin_connected_components_of
closedin_trans_full compact_imp_locally_compact_space)
lemma intermediate_continuum_exists:
assumes "connected_space X" "locally_compact_space X" "Hausdorff_space X"
and C: "compactin X C" "connectedin X C" "C \<noteq> {}" "C \<noteq> topspace X"
and U: "openin X U" "C \<subseteq> U"
obtains D where "compactin X D" "connectedin X D" "C \<subset> D" "D \<subset> U"
proof -
have "C \<subseteq> topspace X"
by (simp add: C compactin_subset_topspace)
with C obtain a where a: "a \<in> topspace X" "a \<notin> C"
by blast
moreover have "compactin (subtopology X (U - {a})) C"
by (simp add: C U a compact_imp_compactin_subtopology subset_Diff_insert)
moreover have "Hausdorff_space (subtopology X (U - {a}))"
using Hausdorff_space_subtopology assms(3) by blast
moreover
have "locally_compact_space (subtopology X (U - {a}))"
by (rule locally_compact_space_open_subset)
(auto simp: locally_compact_Hausdorff_imp_regular_space open_in_Hausdorff_delete assms)
ultimately obtain V K where V: "openin X V" "a \<notin> V" "V \<subseteq> U" and K: "compactin X K" "a \<notin> K" "K \<subseteq> U"
and cloK: "closedin (subtopology X (U - {a})) K" and "C \<subseteq> V" "V \<subseteq> K"
using locally_compact_space_compact_closed_compact [of "subtopology X (U - {a})"] assms
by (smt (verit, del_insts) Diff_empty compactin_subtopology open_in_Hausdorff_delete openin_open_subtopology subset_Diff_insert)
then obtain D where D: "D \<in> connected_components_of (subtopology X K)" and "C \<subseteq> D"
using C
by (metis compactin_subset_topspace connected_component_in_connected_components_of
connected_component_of_maximal connectedin_subtopology subset_empty subset_eq topspace_subtopology_subset)
show thesis
proof
have cloD: "closedin (subtopology X K) D"
by (simp add: D closedin_connected_components_of)
then have XKD: "compactin (subtopology X K) D"
by (simp add: K closedin_compact_space compact_space_subtopology)
then show "compactin X D"
by (simp add: compactin_subtopology)
show "connectedin X D"
using D connectedin_connected_components_of connectedin_subtopology by blast
have "K \<noteq> topspace X"
using K a by blast
moreover have "V \<subseteq> X interior_of K"
by (simp add: \<open>openin X V\<close> \<open>V \<subseteq> K\<close> interior_of_maximal)
ultimately have "C \<noteq> D"
using boundary_bumping_theorem_closed_gen [of X K C] D \<open>C \<subseteq> V\<close>
by (auto simp add: assms K compactin_imp_closedin frontier_of_def)
then show "C \<subset> D"
using \<open>C \<subseteq> D\<close> by blast
have "D \<subseteq> U"
using K(3) \<open>closedin (subtopology X K) D\<close> closedin_imp_subset by blast
moreover have "D \<noteq> U"
using K XKD \<open>C \<subset> D\<close> assms
by (metis \<open>K \<noteq> topspace X\<close> cloD closedin_imp_subset compactin_imp_closedin connected_space_clopen_in
inf_bot_left inf_le2 subset_antisym)
ultimately
show "D \<subset> U" by blast
qed
qed
lemma boundary_bumping_theorem_gen:
assumes X: "connected_space X" "locally_compact_space X" "Hausdorff_space X"
and "S \<subset> topspace X" and C: "C \<in> connected_components_of(subtopology X S)"
and compC: "compactin X (X closure_of C)"
shows "X frontier_of C \<inter> X frontier_of S \<noteq> {}"
proof -
have Csub: "C \<subseteq> topspace X" "C \<subseteq> S" and "connectedin X C"
using C connectedin_connected_components_of connectedin_subset_topspace connectedin_subtopology
by fastforce+
have "C \<noteq> {}"
using C nonempty_connected_components_of by blast
obtain "X interior_of C \<subseteq> X interior_of S" "X closure_of C \<subseteq> X closure_of S"
by (simp add: Csub closure_of_mono interior_of_mono)
moreover have False if "X closure_of C \<subseteq> X interior_of S"
proof -
have "X closure_of C = C"
by (meson C closedin_connected_component_of_subtopology closure_of_eq interior_of_subset order_trans that)
with that have "C \<subseteq> X interior_of S"
by simp
then obtain D where "compactin X D" and "connectedin X D" and "C \<subset> D" and "D \<subset> X interior_of S"
using intermediate_continuum_exists assms \<open>X closure_of C = C\<close> compC Csub
by (metis \<open>C \<noteq> {}\<close> \<open>connectedin X C\<close> openin_interior_of psubsetE)
then have "D \<subseteq> C"
by (metis C \<open>C \<noteq> {}\<close> connected_components_of_maximal connectedin_subtopology disjnt_def inf.orderE interior_of_subset order_trans psubsetE)
then show False
using \<open>C \<subset> D\<close> by blast
qed
ultimately show ?thesis
by (smt (verit, ccfv_SIG) DiffI disjoint_iff_not_equal frontier_of_def subset_eq)
qed
lemma boundary_bumping_theorem:
"\<lbrakk>connected_space X; compact_space X; Hausdorff_space X; S \<subset> topspace X;
C \<in> connected_components_of(subtopology X S)\<rbrakk>
\<Longrightarrow> X frontier_of C \<inter> X frontier_of S \<noteq> {}"
by (simp add: boundary_bumping_theorem_gen closedin_compact_space compact_imp_locally_compact_space)
subsection \<open>Compactly generated spaces (k-spaces)\<close>
text \<open>These don't have to be Hausdorff\<close>
definition k_space where
"k_space X \<equiv>
\<forall>S. S \<subseteq> topspace X \<longrightarrow>
(closedin X S \<longleftrightarrow> (\<forall>K. compactin X K \<longrightarrow> closedin (subtopology X K) (K \<inter> S)))"
lemma k_space:
"k_space X \<longleftrightarrow>
(\<forall>S. S \<subseteq> topspace X \<and>
(\<forall>K. compactin X K \<longrightarrow> closedin (subtopology X K) (K \<inter> S)) \<longrightarrow> closedin X S)"
by (metis closedin_subtopology inf_commute k_space_def)
lemma k_space_open:
"k_space X \<longleftrightarrow>
(\<forall>S. S \<subseteq> topspace X \<and>
(\<forall>K. compactin X K \<longrightarrow> openin (subtopology X K) (K \<inter> S)) \<longrightarrow> openin X S)"
proof -
have "openin X S"
if "k_space X" "S \<subseteq> topspace X"
and "\<forall>K. compactin X K \<longrightarrow> openin (subtopology X K) (K \<inter> S)" for S
using that unfolding k_space openin_closedin_eq
by (metis Diff_Int_distrib2 Diff_subset inf_commute topspace_subtopology)
moreover have "k_space X"
if "\<forall>S. S \<subseteq> topspace X \<and> (\<forall>K. compactin X K \<longrightarrow> openin (subtopology X K) (K \<inter> S)) \<longrightarrow> openin X S"
unfolding k_space openin_closedin_eq
by (simp add: Diff_Int_distrib closedin_def inf_commute that)
ultimately show ?thesis
by blast
qed
lemma k_space_alt:
"k_space X \<longleftrightarrow>
(\<forall>S. S \<subseteq> topspace X
\<longrightarrow> (openin X S \<longleftrightarrow> (\<forall>K. compactin X K \<longrightarrow> openin (subtopology X K) (K \<inter> S))))"
by (meson k_space_open openin_subtopology_Int2)
lemma k_space_quotient_map_image:
assumes q: "quotient_map X Y q" and X: "k_space X"
shows "k_space Y"
unfolding k_space
proof clarify
fix S
assume "S \<subseteq> topspace Y" and S: "\<forall>K. compactin Y K \<longrightarrow> closedin (subtopology Y K) (K \<inter> S)"
then have iff: "closedin X {x \<in> topspace X. q x \<in> S} \<longleftrightarrow> closedin Y S"
using q quotient_map_closedin by fastforce
have "closedin (subtopology X K) (K \<inter> {x \<in> topspace X. q x \<in> S})" if "compactin X K" for K
proof -
have "{x \<in> topspace X. q x \<in> q ` K} \<inter> K = K"
using compactin_subset_topspace that by blast
then have *: "subtopology X K = subtopology (subtopology X {x \<in> topspace X. q x \<in> q ` K}) K"
by (simp add: subtopology_subtopology)
have **: "K \<inter> {x \<in> topspace X. q x \<in> S} =
K \<inter> {x \<in> topspace (subtopology X {x \<in> topspace X. q x \<in> q ` K}). q x \<in> q ` K \<inter> S}"
by auto
have "K \<subseteq> topspace X"
by (simp add: compactin_subset_topspace that)
show ?thesis
unfolding * **
proof (intro closedin_continuous_map_preimage closedin_subtopology_Int_closed)
show "continuous_map (subtopology X {x \<in> topspace X. q x \<in> q ` K}) (subtopology Y (q ` K)) q"
by (auto simp add: continuous_map_in_subtopology continuous_map_from_subtopology q quotient_imp_continuous_map)
show "closedin (subtopology Y (q ` K)) (q ` K \<inter> S)"
by (meson S image_compactin q quotient_imp_continuous_map that)
qed
qed
then have "closedin X {x \<in> topspace X. q x \<in> S}"
by (metis (no_types, lifting) X k_space mem_Collect_eq subsetI)
with iff show "closedin Y S" by simp
qed
lemma k_space_retraction_map_image:
"\<lbrakk>retraction_map X Y r; k_space X\<rbrakk> \<Longrightarrow> k_space Y"
using k_space_quotient_map_image retraction_imp_quotient_map by blast
lemma homeomorphic_k_space:
"X homeomorphic_space Y \<Longrightarrow> k_space X \<longleftrightarrow> k_space Y"
by (meson homeomorphic_map_def homeomorphic_space homeomorphic_space_sym k_space_quotient_map_image)
lemma k_space_perfect_map_image:
"\<lbrakk>k_space X; perfect_map X Y f\<rbrakk> \<Longrightarrow> k_space Y"
using k_space_quotient_map_image perfect_imp_quotient_map by blast
lemma locally_compact_imp_k_space:
assumes "locally_compact_space X"
shows "k_space X"
unfolding k_space
proof clarify
fix S
assume "S \<subseteq> topspace X" and S: "\<forall>K. compactin X K \<longrightarrow> closedin (subtopology X K) (K \<inter> S)"
have False if non: "\<not> (X closure_of S \<subseteq> S)"
proof -
obtain x where "x \<in> X closure_of S" "x \<notin> S"
using non by blast
then have "x \<in> topspace X"
by (simp add: in_closure_of)
then obtain K U where "openin X U" "compactin X K" "x \<in> U" "U \<subseteq> K"
by (meson assms locally_compact_space_def)
then show False
using \<open>x \<in> X closure_of S\<close> openin_Int_closure_of_eq [OF \<open>openin X U\<close>]
by (smt (verit, ccfv_threshold) Int_iff S \<open>x \<notin> S\<close> closedin_Int_closure_of inf.orderE inf_assoc)
qed
then show "closedin X S"
using S \<open>S \<subseteq> topspace X\<close> closure_of_subset_eq by blast
qed
lemma compact_imp_k_space:
"compact_space X \<Longrightarrow> k_space X"
by (simp add: compact_imp_locally_compact_space locally_compact_imp_k_space)
lemma k_space_discrete_topology: "k_space(discrete_topology U)"
by (simp add: k_space_open)
lemma k_space_closed_subtopology:
assumes "k_space X" "closedin X C"
shows "k_space (subtopology X C)"
unfolding k_space compactin_subtopology
proof clarsimp
fix S
assume Ssub: "S \<subseteq> topspace X" "S \<subseteq> C"
and S: "\<forall>K. compactin X K \<and> K \<subseteq> C \<longrightarrow> closedin (subtopology (subtopology X C) K) (K \<inter> S)"
have "closedin (subtopology X K) (K \<inter> S)" if "compactin X K" for K
proof -
have "closedin (subtopology (subtopology X C) (K \<inter> C)) ((K \<inter> C) \<inter> S)"
by (simp add: S \<open>closedin X C\<close> compact_Int_closedin that)
then show ?thesis
using \<open>closedin X C\<close> Ssub by (auto simp add: closedin_subtopology)
qed
then show "closedin (subtopology X C) S"
by (metis Ssub \<open>k_space X\<close> closedin_subset_topspace k_space_def)
qed
lemma k_space_subtopology:
assumes 1: "\<And>T. \<lbrakk>T \<subseteq> topspace X; T \<subseteq> S;
\<And>K. compactin X K \<Longrightarrow> closedin (subtopology X (K \<inter> S)) (K \<inter> T)\<rbrakk> \<Longrightarrow> closedin (subtopology X S) T"
assumes 2: "\<And>K. compactin X K \<Longrightarrow> k_space(subtopology X (K \<inter> S))"
shows "k_space (subtopology X S)"
unfolding k_space
proof (intro conjI strip)
fix U
assume \<section>: "U \<subseteq> topspace (subtopology X S) \<and> (\<forall>K. compactin (subtopology X S) K \<longrightarrow> closedin (subtopology (subtopology X S) K) (K \<inter> U))"
have "closedin (subtopology X (K \<inter> S)) (K \<inter> U)" if "compactin X K" for K
proof -
have "K \<inter> U \<subseteq> topspace (subtopology X (K \<inter> S))"
using "\<section>" by auto
moreover
have "\<And>K'. compactin (subtopology X (K \<inter> S)) K' \<Longrightarrow> closedin (subtopology (subtopology X (K \<inter> S)) K') (K' \<inter> K \<inter> U)"
by (metis "\<section>" compactin_subtopology inf.orderE inf_commute subtopology_subtopology)
ultimately show ?thesis
by (metis (no_types, opaque_lifting) "2" inf.assoc k_space_def that)
qed
then show "closedin (subtopology X S) U"
using "1" \<section> by auto
qed
lemma k_space_subtopology_open:
assumes 1: "\<And>T. \<lbrakk>T \<subseteq> topspace X; T \<subseteq> S;
\<And>K. compactin X K \<Longrightarrow> openin (subtopology X (K \<inter> S)) (K \<inter> T)\<rbrakk> \<Longrightarrow> openin (subtopology X S) T"
assumes 2: "\<And>K. compactin X K \<Longrightarrow> k_space(subtopology X (K \<inter> S))"
shows "k_space (subtopology X S)"
unfolding k_space_open
proof (intro conjI strip)
fix U
assume \<section>: "U \<subseteq> topspace (subtopology X S) \<and> (\<forall>K. compactin (subtopology X S) K \<longrightarrow> openin (subtopology (subtopology X S) K) (K \<inter> U))"
have "openin (subtopology X (K \<inter> S)) (K \<inter> U)" if "compactin X K" for K
proof -
have "K \<inter> U \<subseteq> topspace (subtopology X (K \<inter> S))"
using "\<section>" by auto
moreover
have "\<And>K'. compactin (subtopology X (K \<inter> S)) K' \<Longrightarrow> openin (subtopology (subtopology X (K \<inter> S)) K') (K' \<inter> K \<inter> U)"
by (metis "\<section>" compactin_subtopology inf.orderE inf_commute subtopology_subtopology)
ultimately show ?thesis
by (metis (no_types, opaque_lifting) "2" inf.assoc k_space_open that)
qed
then show "openin (subtopology X S) U"
using "1" \<section> by auto
qed
lemma k_space_open_subtopology_aux:
assumes "kc_space X" "compact_space X" "openin X V"
shows "k_space (subtopology X V)"
proof (clarsimp simp: k_space subtopology_subtopology compactin_subtopology Int_absorb1)
fix S
assume "S \<subseteq> topspace X"
and "S \<subseteq> V"
and S: "\<forall>K. compactin X K \<and> K \<subseteq> V \<longrightarrow> closedin (subtopology X K) (K \<inter> S)"
then have "V \<subseteq> topspace X"
using assms openin_subset by blast
have "S = V \<inter> ((topspace X - V) \<union> S)"
using \<open>S \<subseteq> V\<close> by auto
moreover have "closedin (subtopology X V) (V \<inter> ((topspace X - V) \<union> S))"
proof (intro closedin_subtopology_Int_closed compactin_imp_closedin_gen \<open>kc_space X\<close>)
show "compactin X (topspace X - V \<union> S)"
unfolding compactin_def
proof (intro conjI strip)
show "topspace X - V \<union> S \<subseteq> topspace X"
by (simp add: \<open>S \<subseteq> topspace X\<close>)
fix \<U>
assume \<U>: "Ball \<U> (openin X) \<and> topspace X - V \<union> S \<subseteq> \<Union>\<U>"
moreover
have "compactin X (topspace X - V)"
using assms closedin_compact_space by blast
ultimately obtain \<G> where "finite \<G>" "\<G> \<subseteq> \<U>" and \<G>: "topspace X - V \<subseteq> \<Union>\<G>"
unfolding compactin_def using \<open>V \<subseteq> topspace X\<close> by (metis le_sup_iff)
then have "topspace X - \<Union>\<G> \<subseteq> V"
by blast
then have "closedin (subtopology X (topspace X - \<Union>\<G>)) ((topspace X - \<Union>\<G>) \<inter> S)"
by (meson S \<U> \<open>\<G> \<subseteq> \<U>\<close> \<open>compact_space X\<close> closedin_compact_space openin_Union openin_closedin_eq subset_iff)
then have "compactin X ((topspace X - \<Union>\<G>) \<inter> S)"
by (meson \<U> \<open>\<G> \<subseteq> \<U>\<close>\<open>compact_space X\<close> closedin_compact_space closedin_trans_full openin_Union openin_closedin_eq subset_iff)
then obtain \<H> where "finite \<H>" "\<H> \<subseteq> \<U>" "(topspace X - \<Union>\<G>) \<inter> S \<subseteq> \<Union>\<H>"
unfolding compactin_def by (smt (verit, best) \<U> inf_le2 subset_trans sup.boundedE)
with \<G> have "topspace X - V \<union> S \<subseteq> \<Union>(\<G> \<union> \<H>)"
using \<open>S \<subseteq> topspace X\<close> by auto
then show "\<exists>\<F>. finite \<F> \<and> \<F> \<subseteq> \<U> \<and> topspace X - V \<union> S \<subseteq> \<Union>\<F>"
by (metis \<open>\<G> \<subseteq> \<U>\<close> \<open>\<H> \<subseteq> \<U>\<close> \<open>finite \<G>\<close> \<open>finite \<H>\<close> finite_Un le_sup_iff)
qed
qed
ultimately show "closedin (subtopology X V) S"
by metis
qed
lemma k_space_open_subtopology:
assumes X: "kc_space X \<or> Hausdorff_space X \<or> regular_space X" and "k_space X" "openin X S"
shows "k_space(subtopology X S)"
proof (rule k_space_subtopology_open)
fix T
assume "T \<subseteq> topspace X"
and "T \<subseteq> S"
and T: "\<And>K. compactin X K \<Longrightarrow> openin (subtopology X (K \<inter> S)) (K \<inter> T)"
have "openin (subtopology X K) (K \<inter> T)" if "compactin X K" for K
by (smt (verit, ccfv_threshold) T assms(3) inf_assoc inf_commute openin_Int openin_subtopology that)
then show "openin (subtopology X S) T"
by (metis \<open>T \<subseteq> S\<close> \<open>T \<subseteq> topspace X\<close> assms(2) k_space_alt subset_openin_subtopology)
next
fix K
assume "compactin X K"
then have KS: "openin (subtopology X K) (K \<inter> S)"
by (simp add: \<open>openin X S\<close> openin_subtopology_Int2)
have XK: "compact_space (subtopology X K)"
by (simp add: \<open>compactin X K\<close> compact_space_subtopology)
show "k_space (subtopology X (K \<inter> S))"
using X
proof (rule disjE)
assume "kc_space X"
then show "k_space (subtopology X (K \<inter> S))"
using k_space_open_subtopology_aux [of "subtopology X K" "K \<inter> S"]
by (simp add: KS XK kc_space_subtopology subtopology_subtopology)
next
assume "Hausdorff_space X \<or> regular_space X"
then have "locally_compact_space (subtopology (subtopology X K) (K \<inter> S))"
using locally_compact_space_open_subset Hausdorff_space_subtopology KS XK
compact_imp_locally_compact_space regular_space_subtopology by blast
then show "k_space (subtopology X (K \<inter> S))"
by (simp add: locally_compact_imp_k_space subtopology_subtopology)
qed
qed
lemma k_kc_space_subtopology:
"\<lbrakk>k_space X; kc_space X; openin X S \<or> closedin X S\<rbrakk> \<Longrightarrow> k_space(subtopology X S) \<and> kc_space(subtopology X S)"
by (metis k_space_closed_subtopology k_space_open_subtopology kc_space_subtopology)
lemma k_space_as_quotient_explicit:
"k_space X \<longleftrightarrow> quotient_map (sum_topology (subtopology X) {K. compactin X K}) X snd"
proof -
have [simp]: "{x \<in> topspace X. x \<in> K \<and> x \<in> U} = K \<inter> U" if "U \<subseteq> topspace X" for K U
using that by blast
show "?thesis"
apply (simp add: quotient_map_def openin_sum_topology snd_image_Sigma k_space_alt)
by (smt (verit, del_insts) Union_iff compactin_sing inf.orderE mem_Collect_eq singletonI subsetI)
qed
lemma k_space_as_quotient:
fixes X :: "'a topology"
shows "k_space X \<longleftrightarrow> (\<exists>q. \<exists>Y:: ('a set * 'a) topology. locally_compact_space Y \<and> quotient_map Y X q)"
(is "?lhs=?rhs")
proof
show "k_space X" if ?rhs
using that k_space_quotient_map_image locally_compact_imp_k_space by blast
next
assume "k_space X"
show ?rhs
proof (intro exI conjI)
show "locally_compact_space (sum_topology (subtopology X) {K. compactin X K})"
by (simp add: compact_imp_locally_compact_space compact_space_subtopology locally_compact_space_sum_topology)
show "quotient_map (sum_topology (subtopology X) {K. compactin X K}) X snd"
using \<open>k_space X\<close> k_space_as_quotient_explicit by blast
qed
qed
lemma k_space_prod_topology_left:
assumes X: "locally_compact_space X" "Hausdorff_space X \<or> regular_space X" and "k_space Y"
shows "k_space (prod_topology X Y)"
proof -
obtain q and Z :: "('b set * 'b) topology" where "locally_compact_space Z" and q: "quotient_map Z Y q"
using \<open>k_space Y\<close> k_space_as_quotient by blast
then show ?thesis
using quotient_map_prod_right [OF X q] X k_space_quotient_map_image locally_compact_imp_k_space
locally_compact_space_prod_topology by blast
qed
lemma k_space_prod_topology_right:
assumes "k_space X" and Y: "locally_compact_space Y" "Hausdorff_space Y \<or> regular_space Y"
shows "k_space (prod_topology X Y)"
using assms homeomorphic_k_space homeomorphic_space_prod_topology_swap k_space_prod_topology_left by blast
lemma continuous_map_from_k_space:
assumes "k_space X" and f: "\<And>K. compactin X K \<Longrightarrow> continuous_map(subtopology X K) Y f"
shows "continuous_map X Y f"
proof -
have "\<And>x. x \<in> topspace X \<Longrightarrow> f x \<in> topspace Y"
by (metis compactin_absolute compactin_sing f image_compactin image_empty image_insert)
moreover have "closedin X {x \<in> topspace X. f x \<in> C}" if "closedin Y C" for C
proof -
have "{x \<in> topspace X. f x \<in> C} \<subseteq> topspace X"
by fastforce
moreover
have eq: "K \<inter> {x \<in> topspace X. f x \<in> C} = {x. x \<in> topspace(subtopology X K) \<and> f x \<in> (f ` K \<inter> C)}" for K
by auto
have "closedin (subtopology X K) (K \<inter> {x \<in> topspace X. f x \<in> C})" if "compactin X K" for K
unfolding eq
proof (rule closedin_continuous_map_preimage)
show "continuous_map (subtopology X K) (subtopology Y (f`K)) f"
by (simp add: continuous_map_in_subtopology f image_mono that)
show "closedin (subtopology Y (f`K)) (f ` K \<inter> C)"
using \<open>closedin Y C\<close> closedin_subtopology by blast
qed
ultimately show ?thesis
using \<open>k_space X\<close> k_space by blast
qed
ultimately show ?thesis
by (simp add: continuous_map_closedin)
qed
lemma closed_map_into_k_space:
assumes "k_space Y" and fim: "f \<in> (topspace X) \<rightarrow> topspace Y"
and f: "\<And>K. compactin Y K
\<Longrightarrow> closed_map(subtopology X {x \<in> topspace X. f x \<in> K}) (subtopology Y K) f"
shows "closed_map X Y f"
unfolding closed_map_def
proof (intro strip)
fix C
assume "closedin X C"
have "closedin (subtopology Y K) (K \<inter> f ` C)"
if "compactin Y K" for K
proof -
have eq: "K \<inter> f ` C = f ` ({x \<in> topspace X. f x \<in> K} \<inter> C)"
using \<open>closedin X C\<close> closedin_subset by auto
show ?thesis
unfolding eq
by (metis (no_types, lifting) \<open>closedin X C\<close> closed_map_def closedin_subtopology f inf_commute that)
qed
then show "closedin Y (f ` C)"
using \<open>k_space Y\<close> unfolding k_space
by (meson \<open>closedin X C\<close> closedin_subset order.trans fim funcset_image subset_image_iff)
qed
text \<open>Essentially the same proof\<close>
lemma open_map_into_k_space:
assumes "k_space Y" and fim: "f \<in> (topspace X) \<rightarrow> topspace Y"
and f: "\<And>K. compactin Y K
\<Longrightarrow> open_map (subtopology X {x \<in> topspace X. f x \<in> K}) (subtopology Y K) f"
shows "open_map X Y f"
unfolding open_map_def
proof (intro strip)
fix C
assume "openin X C"
have "openin (subtopology Y K) (K \<inter> f ` C)"
if "compactin Y K" for K
proof -
have eq: "K \<inter> f ` C = f ` ({x \<in> topspace X. f x \<in> K} \<inter> C)"
using \<open>openin X C\<close> openin_subset by auto
show ?thesis
unfolding eq
by (metis (no_types, lifting) \<open>openin X C\<close> open_map_def openin_subtopology f inf_commute that)
qed
then show "openin Y (f ` C)"
using \<open>k_space Y\<close> unfolding k_space_open
by (meson \<open>openin X C\<close> openin_subset order.trans fim funcset_image subset_image_iff)
qed
lemma quotient_map_into_k_space:
fixes f :: "'a \<Rightarrow> 'b"
assumes "k_space Y" and cmf: "continuous_map X Y f"
and fim: "f ` (topspace X) = topspace Y"
and f: "\<And>k. compactin Y k
\<Longrightarrow> quotient_map (subtopology X {x \<in> topspace X. f x \<in> k})
(subtopology Y k) f"
shows "quotient_map X Y f"
proof -
have "closedin Y C"
if "C \<subseteq> topspace Y" and K: "closedin X {x \<in> topspace X. f x \<in> C}" for C
proof -
have "closedin (subtopology Y K) (K \<inter> C)" if "compactin Y K" for K
proof -
define Kf where "Kf \<equiv> {x \<in> topspace X. f x \<in> K}"
have *: "K \<inter> C \<subseteq> topspace Y \<and> K \<inter> C \<subseteq> K"
using \<open>C \<subseteq> topspace Y\<close> by blast
then have eq: "closedin (subtopology X Kf) (Kf \<inter> {x \<in> topspace X. f x \<in> C}) =
closedin (subtopology Y K) (K \<inter> C)"
using f [OF that] * unfolding quotient_map_closedin Kf_def
by (smt (verit, ccfv_SIG) Collect_cong Int_def compactin_subset_topspace mem_Collect_eq that topspace_subtopology topspace_subtopology_subset)
have dd: "{x \<in> topspace X \<inter> Kf. f x \<in> K \<inter> C} = Kf \<inter> {x \<in> topspace X. f x \<in> C}"
by (auto simp add: Kf_def)
have "closedin (subtopology X Kf) {x \<in> topspace X. x \<in> Kf \<and> f x \<in> K \<and> f x \<in> C}"
using K closedin_subtopology by (fastforce simp add: Kf_def)
with K closedin_subtopology_Int_closed eq show ?thesis
by blast
qed
then show ?thesis
using \<open>k_space Y\<close> that unfolding k_space by blast
qed
moreover have "closedin X {x \<in> topspace X. f x \<in> K}"
if "K \<subseteq> topspace Y" "closedin Y K" for K
using that cmf continuous_map_closedin by fastforce
ultimately show ?thesis
unfolding quotient_map_closedin using fim by blast
qed
lemma quotient_map_into_k_space_eq:
assumes "k_space Y" "kc_space Y"
shows "quotient_map X Y f \<longleftrightarrow>
continuous_map X Y f \<and> f ` (topspace X) = topspace Y \<and>
(\<forall>K. compactin Y K
\<longrightarrow> quotient_map (subtopology X {x \<in> topspace X. f x \<in> K}) (subtopology Y K) f)"
using assms
by (auto simp: kc_space_def intro: quotient_map_into_k_space quotient_map_restriction
dest: quotient_imp_continuous_map quotient_imp_surjective_map)
lemma open_map_into_k_space_eq:
assumes "k_space Y"
shows "open_map X Y f \<longleftrightarrow>
f \<in> (topspace X) \<rightarrow> topspace Y \<and>
(\<forall>k. compactin Y k
\<longrightarrow> open_map (subtopology X {x \<in> topspace X. f x \<in> k}) (subtopology Y k) f)"
using assms open_map_imp_subset_topspace open_map_into_k_space open_map_restriction by fastforce
lemma closed_map_into_k_space_eq:
assumes "k_space Y"
shows "closed_map X Y f \<longleftrightarrow>
f \<in> (topspace X) \<rightarrow> topspace Y \<and>
(\<forall>k. compactin Y k
\<longrightarrow> closed_map (subtopology X {x \<in> topspace X. f x \<in> k}) (subtopology Y k) f)"
(is "?lhs \<longleftrightarrow> ?rhs")
proof
show "?lhs \<Longrightarrow> ?rhs"
by (simp add: closed_map_imp_subset_topspace closed_map_restriction)
show "?rhs \<Longrightarrow> ?lhs"
by (simp add: assms closed_map_into_k_space)
qed
lemma proper_map_into_k_space:
assumes "k_space Y" and fim: "f \<in> (topspace X) \<rightarrow> topspace Y"
and f: "\<And>K. compactin Y K
\<Longrightarrow> proper_map (subtopology X {x \<in> topspace X. f x \<in> K})
(subtopology Y K) f"
shows "proper_map X Y f"
proof -
have "closed_map X Y f"
by (meson assms closed_map_into_k_space fim proper_map_def)
with f topspace_subtopology_subset show ?thesis
apply (simp add: proper_map_alt)
by (smt (verit, best) Collect_cong compactin_absolute)
qed
lemma proper_map_into_k_space_eq:
assumes "k_space Y"
shows "proper_map X Y f \<longleftrightarrow>
f \<in> (topspace X) \<rightarrow> topspace Y \<and>
(\<forall>K. compactin Y K
\<longrightarrow> proper_map (subtopology X {x \<in> topspace X. f x \<in> K}) (subtopology Y K) f)"
(is "?lhs \<longleftrightarrow> ?rhs")
proof
show "?lhs \<Longrightarrow> ?rhs"
by (simp add: proper_map_imp_subset_topspace proper_map_restriction)
show "?rhs \<Longrightarrow> ?lhs"
by (simp add: assms funcset_image proper_map_into_k_space)
qed
lemma compact_imp_proper_map:
assumes "k_space Y" "kc_space Y" and fim: "f \<in> (topspace X) \<rightarrow> topspace Y"
and f: "continuous_map X Y f \<or> kc_space X"
and comp: "\<And>K. compactin Y K \<Longrightarrow> compactin X {x \<in> topspace X. f x \<in> K}"
shows "proper_map X Y f"
proof (rule compact_imp_proper_map_gen)
fix S
assume "S \<subseteq> topspace Y"
and "\<And>K. compactin Y K \<Longrightarrow> compactin Y (S \<inter> K)"
with assms show "closedin Y S"
by (simp add: closedin_subset_topspace inf_commute k_space kc_space_def)
qed (use assms in auto)
lemma proper_eq_compact_map:
assumes "k_space Y" "kc_space Y"
and f: "continuous_map X Y f \<or> kc_space X"
shows "proper_map X Y f \<longleftrightarrow>
f \<in> (topspace X) \<rightarrow> topspace Y \<and>
(\<forall>K. compactin Y K \<longrightarrow> compactin X {x \<in> topspace X. f x \<in> K})"
(is "?lhs \<longleftrightarrow> ?rhs")
proof
show "?lhs \<Longrightarrow> ?rhs"
using \<open>k_space Y\<close> compactin_proper_map_preimage proper_map_into_k_space_eq by blast
qed (use assms compact_imp_proper_map in auto)
lemma compact_imp_perfect_map:
assumes "k_space Y" "kc_space Y" and "f ` (topspace X) = topspace Y"
and "continuous_map X Y f"
and "\<And>K. compactin Y K \<Longrightarrow> compactin X {x \<in> topspace X. f x \<in> K}"
shows "perfect_map X Y f"
by (simp add: assms compact_imp_proper_map perfect_map_def flip: image_subset_iff_funcset)
end