section \<open>Neighbourhood bases and Locally path-connected spaces\<close>
theory Locally
imports
Path_Connected Function_Topology Sum_Topology
begin
subsection\<open>Neighbourhood Bases\<close>
text \<open>Useful for "local" properties of various kinds\<close>
definition neighbourhood_base_at where
"neighbourhood_base_at x P X \<equiv>
\<forall>W. openin X 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)"
definition neighbourhood_base_of where
"neighbourhood_base_of P X \<equiv>
\<forall>x \<in> topspace X. neighbourhood_base_at x P X"
lemma neighbourhood_base_of:
"neighbourhood_base_of P X \<longleftrightarrow>
(\<forall>W x. openin X 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))"
unfolding neighbourhood_base_at_def neighbourhood_base_of_def
using openin_subset by blast
lemma neighbourhood_base_at_mono:
"\<lbrakk>neighbourhood_base_at x P X; \<And>S. \<lbrakk>P S; x \<in> S\<rbrakk> \<Longrightarrow> Q S\<rbrakk> \<Longrightarrow> neighbourhood_base_at x Q X"
unfolding neighbourhood_base_at_def
by (meson subset_eq)
lemma neighbourhood_base_of_mono:
"\<lbrakk>neighbourhood_base_of P X; \<And>S. P S \<Longrightarrow> Q S\<rbrakk> \<Longrightarrow> neighbourhood_base_of Q X"
unfolding neighbourhood_base_of_def
using neighbourhood_base_at_mono by force
lemma open_neighbourhood_base_at:
"(\<And>S. \<lbrakk>P S; x \<in> S\<rbrakk> \<Longrightarrow> openin X S)
\<Longrightarrow> neighbourhood_base_at x P X \<longleftrightarrow> (\<forall>W. openin X W \<and> x \<in> W \<longrightarrow> (\<exists>U. P U \<and> x \<in> U \<and> U \<subseteq> W))"
unfolding neighbourhood_base_at_def
by (meson subsetD)
lemma open_neighbourhood_base_of:
"(\<forall>S. P S \<longrightarrow> openin X S)
\<Longrightarrow> neighbourhood_base_of P X \<longleftrightarrow> (\<forall>W x. openin X W \<and> x \<in> W \<longrightarrow> (\<exists>U. P U \<and> x \<in> U \<and> U \<subseteq> W))"
apply (simp add: neighbourhood_base_of, safe, blast)
by meson
lemma neighbourhood_base_of_open_subset:
"\<lbrakk>neighbourhood_base_of P X; openin X S\<rbrakk>
\<Longrightarrow> neighbourhood_base_of P (subtopology X S)"
apply (clarsimp simp add: neighbourhood_base_of openin_subtopology_alt image_def)
apply (rename_tac x V)
apply (drule_tac x="S \<inter> V" in spec)
apply (simp add: Int_ac)
by (metis IntI le_infI1 openin_Int)
lemma neighbourhood_base_of_topology_base:
"openin X = arbitrary union_of (\<lambda>W. W \<in> \<B>)
\<Longrightarrow> neighbourhood_base_of P X \<longleftrightarrow>
(\<forall>W x. W \<in> \<B> \<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 (auto simp: openin_topology_base_unique neighbourhood_base_of)
by (meson subset_trans)
lemma neighbourhood_base_at_unlocalized:
assumes "\<And>S T. \<lbrakk>P S; openin X T; x \<in> T; T \<subseteq> S\<rbrakk> \<Longrightarrow> P T"
shows "neighbourhood_base_at x P X
\<longleftrightarrow> (x \<in> topspace X \<longrightarrow> (\<exists>U V. openin X U \<and> P V \<and> x \<in> U \<and> U \<subseteq> V \<and> V \<subseteq> topspace X))"
(is "?lhs = ?rhs")
proof
assume R: ?rhs show ?lhs
unfolding neighbourhood_base_at_def
proof clarify
fix W
assume "openin X W" "x \<in> W"
then have "x \<in> topspace X"
using openin_subset by blast
with R obtain U V where "openin X U" "P V" "x \<in> U" "U \<subseteq> V" "V \<subseteq> topspace X"
by blast
then show "\<exists>U V. openin X U \<and> P V \<and> x \<in> U \<and> U \<subseteq> V \<and> V \<subseteq> W"
by (metis IntI \<open>openin X W\<close> \<open>x \<in> W\<close> assms inf_le1 inf_le2 openin_Int)
qed
qed (simp add: neighbourhood_base_at_def)
lemma neighbourhood_base_at_with_subset:
"\<lbrakk>openin X U; x \<in> U\<rbrakk>
\<Longrightarrow> (neighbourhood_base_at x P X \<longleftrightarrow> neighbourhood_base_at x (\<lambda>T. T \<subseteq> U \<and> P T) X)"
apply (simp add: neighbourhood_base_at_def)
apply (metis IntI Int_subset_iff openin_Int)
done
lemma neighbourhood_base_of_with_subset:
"neighbourhood_base_of P X \<longleftrightarrow> neighbourhood_base_of (\<lambda>t. t \<subseteq> topspace X \<and> P t) X"
using neighbourhood_base_at_with_subset
by (fastforce simp add: neighbourhood_base_of_def)
subsection\<open>Locally path-connected spaces\<close>
definition weakly_locally_path_connected_at
where "weakly_locally_path_connected_at x X \<equiv> neighbourhood_base_at x (path_connectedin X) X"
definition locally_path_connected_at
where "locally_path_connected_at x X \<equiv>
neighbourhood_base_at x (\<lambda>U. openin X U \<and> path_connectedin X U) X"
definition locally_path_connected_space
where "locally_path_connected_space X \<equiv> neighbourhood_base_of (path_connectedin X) X"
lemma locally_path_connected_space_alt:
"locally_path_connected_space X \<longleftrightarrow> neighbourhood_base_of (\<lambda>U. openin X U \<and> path_connectedin X U) X"
(is "?P = ?Q")
and locally_path_connected_space_eq_open_path_component_of:
"locally_path_connected_space X \<longleftrightarrow>
(\<forall>U x. openin X U \<and> x \<in> U \<longrightarrow> openin X (Collect (path_component_of (subtopology X U) x)))"
(is "?P = ?R")
proof -
have ?P if ?Q
using locally_path_connected_space_def neighbourhood_base_of_mono that by auto
moreover have ?R if P: ?P
proof -
show ?thesis
proof clarify
fix U y
assume "openin X U" "y \<in> U"
have "\<exists>T. openin X T \<and> x \<in> T \<and> T \<subseteq> Collect (path_component_of (subtopology X U) y)"
if "path_component_of (subtopology X U) y x" for x
proof -
have "x \<in> U"
using path_component_of_equiv that topspace_subtopology by fastforce
then have "\<exists>Ua. openin X Ua \<and> (\<exists>V. path_connectedin X V \<and> x \<in> Ua \<and> Ua \<subseteq> V \<and> V \<subseteq> U)"
using P
by (simp add: \<open>openin X U\<close> locally_path_connected_space_def neighbourhood_base_of)
then show ?thesis
by (metis dual_order.trans path_component_of_equiv path_component_of_maximal path_connectedin_subtopology subset_iff that)
qed
then show "openin X (Collect (path_component_of (subtopology X U) y))"
using openin_subopen by force
qed
qed
moreover have ?Q if ?R
using that
apply (simp add: open_neighbourhood_base_of)
by (metis mem_Collect_eq openin_subset path_component_of_refl path_connectedin_path_component_of path_connectedin_subtopology that topspace_subtopology_subset)
ultimately show "?P = ?Q" "?P = ?R"
by blast+
qed
lemma locally_path_connected_space:
"locally_path_connected_space X
\<longleftrightarrow> (\<forall>V x. openin X V \<and> x \<in> V \<longrightarrow> (\<exists>U. openin X U \<and> path_connectedin X U \<and> x \<in> U \<and> U \<subseteq> V))"
by (simp add: locally_path_connected_space_alt open_neighbourhood_base_of)
lemma locally_path_connected_space_open_path_components:
"locally_path_connected_space X \<longleftrightarrow>
(\<forall>U c. openin X U \<and> c \<in> path_components_of(subtopology X U) \<longrightarrow> openin X c)"
apply (auto simp: locally_path_connected_space_eq_open_path_component_of path_components_of_def)
by (metis imageI inf.absorb_iff2 openin_closedin_eq)
lemma openin_path_component_of_locally_path_connected_space:
"locally_path_connected_space X \<Longrightarrow> openin X (Collect (path_component_of X x))"
apply (auto simp: locally_path_connected_space_eq_open_path_component_of)
by (metis openin_empty openin_topspace path_component_of_eq_empty subtopology_topspace)
lemma openin_path_components_of_locally_path_connected_space:
"\<lbrakk>locally_path_connected_space X; c \<in> path_components_of X\<rbrakk> \<Longrightarrow> openin X c"
apply (auto simp: locally_path_connected_space_eq_open_path_component_of)
by (metis (no_types, lifting) imageE openin_topspace path_components_of_def subtopology_topspace)
lemma closedin_path_components_of_locally_path_connected_space:
"\<lbrakk>locally_path_connected_space X; c \<in> path_components_of X\<rbrakk> \<Longrightarrow> closedin X c"
by (simp add: closedin_def complement_path_components_of_Union openin_path_components_of_locally_path_connected_space openin_clauses(3) path_components_of_subset)
lemma closedin_path_component_of_locally_path_connected_space:
assumes "locally_path_connected_space X"
shows "closedin X (Collect (path_component_of X x))"
proof (cases "x \<in> topspace X")
case True
then show ?thesis
by (simp add: assms closedin_path_components_of_locally_path_connected_space path_component_in_path_components_of)
next
case False
then show ?thesis
by (metis closedin_empty path_component_of_eq_empty)
qed
lemma weakly_locally_path_connected_at:
"weakly_locally_path_connected_at x X \<longleftrightarrow>
(\<forall>V. openin X V \<and> x \<in> V
\<longrightarrow> (\<exists>U. openin X U \<and> x \<in> U \<and> U \<subseteq> V \<and>
(\<forall>y \<in> U. \<exists>C. path_connectedin X C \<and> C \<subseteq> V \<and> x \<in> C \<and> y \<in> C)))"
(is "?lhs = ?rhs")
proof
assume ?lhs then show ?rhs
apply (simp add: weakly_locally_path_connected_at_def neighbourhood_base_at_def)
by (meson order_trans subsetD)
next
have *: "\<exists>V. path_connectedin X V \<and> U \<subseteq> V \<and> V \<subseteq> W"
if "(\<forall>y\<in>U. \<exists>C. path_connectedin X C \<and> C \<subseteq> W \<and> x \<in> C \<and> y \<in> C)"
for W U
proof (intro exI conjI)
let ?V = "(Collect (path_component_of (subtopology X W) x))"
show "path_connectedin X (Collect (path_component_of (subtopology X W) x))"
by (meson path_connectedin_path_component_of path_connectedin_subtopology)
show "U \<subseteq> ?V"
by (auto simp: path_component_of path_connectedin_subtopology that)
show "?V \<subseteq> W"
by (meson path_connectedin_path_component_of path_connectedin_subtopology)
qed
assume ?rhs
then show ?lhs
unfolding weakly_locally_path_connected_at_def neighbourhood_base_at_def by (metis "*")
qed
lemma locally_path_connected_space_im_kleinen:
"locally_path_connected_space X \<longleftrightarrow>
(\<forall>V x. openin X V \<and> x \<in> V
\<longrightarrow> (\<exists>U. openin X U \<and>
x \<in> U \<and> U \<subseteq> V \<and>
(\<forall>y \<in> U. \<exists>c. path_connectedin X c \<and>
c \<subseteq> V \<and> x \<in> c \<and> y \<in> c)))"
apply (simp add: locally_path_connected_space_def neighbourhood_base_of_def)
apply (simp add: weakly_locally_path_connected_at flip: weakly_locally_path_connected_at_def)
using openin_subset apply force
done
lemma locally_path_connected_space_open_subset:
"\<lbrakk>locally_path_connected_space X; openin X s\<rbrakk>
\<Longrightarrow> locally_path_connected_space (subtopology X s)"
apply (simp add: locally_path_connected_space_def neighbourhood_base_of openin_open_subtopology path_connectedin_subtopology)
by (meson order_trans)
lemma locally_path_connected_space_quotient_map_image:
assumes f: "quotient_map X Y f" and X: "locally_path_connected_space X"
shows "locally_path_connected_space Y"
unfolding locally_path_connected_space_open_path_components
proof clarify
fix V C
assume V: "openin Y V" and c: "C \<in> path_components_of (subtopology Y V)"
then have sub: "C \<subseteq> topspace Y"
using path_connectedin_path_components_of path_connectedin_subset_topspace path_connectedin_subtopology by blast
have "openin X {x \<in> topspace X. f x \<in> C}"
proof (subst openin_subopen, clarify)
fix x
assume x: "x \<in> topspace X" and "f x \<in> C"
let ?T = "Collect (path_component_of (subtopology X {z \<in> topspace X. f z \<in> V}) x)"
show "\<exists>T. openin X T \<and> x \<in> T \<and> T \<subseteq> {x \<in> topspace X. f x \<in> C}"
proof (intro exI conjI)
have "\<exists>U. openin X U \<and> ?T \<in> path_components_of (subtopology X U)"
proof (intro exI conjI)
show "openin X ({z \<in> topspace X. f z \<in> V})"
using V f openin_subset quotient_map_def by fastforce
show "Collect (path_component_of (subtopology X {z \<in> topspace X. f z \<in> V}) x)
\<in> path_components_of (subtopology X {z \<in> topspace X. f z \<in> V})"
by (metis (no_types, lifting) Int_iff \<open>f x \<in> C\<close> c mem_Collect_eq path_component_in_path_components_of path_components_of_subset topspace_subtopology topspace_subtopology_subset x)
qed
with X show "openin X ?T"
using locally_path_connected_space_open_path_components by blast
show x: "x \<in> ?T"
using V \<open>f x \<in> C\<close> c openin_subset path_component_of_equiv path_components_of_subset topspace_subtopology topspace_subtopology_subset x
by fastforce
have "f ` ?T \<subseteq> C"
proof (rule path_components_of_maximal)
show "C \<in> path_components_of (subtopology Y V)"
by (simp add: c)
show "path_connectedin (subtopology Y V) (f ` ?T)"
proof -
have "quotient_map (subtopology X {a \<in> topspace X. f a \<in> V}) (subtopology Y V) f"
by (simp add: V f quotient_map_restriction)
then show ?thesis
by (meson path_connectedin_continuous_map_image path_connectedin_path_component_of quotient_imp_continuous_map)
qed
show "\<not> disjnt C (f ` ?T)"
by (metis (no_types, lifting) \<open>f x \<in> C\<close> x disjnt_iff image_eqI)
qed
then show "?T \<subseteq> {x \<in> topspace X. f x \<in> C}"
by (force simp: path_component_of_equiv)
qed
qed
then show "openin Y C"
using f sub by (simp add: quotient_map_def)
qed
lemma homeomorphic_locally_path_connected_space:
assumes "X homeomorphic_space Y"
shows "locally_path_connected_space X \<longleftrightarrow> locally_path_connected_space Y"
proof -
obtain f g where "homeomorphic_maps X Y f g"
using assms homeomorphic_space_def by fastforce
then show ?thesis
by (metis (no_types) homeomorphic_map_def homeomorphic_maps_map locally_path_connected_space_quotient_map_image)
qed
lemma locally_path_connected_space_retraction_map_image:
"\<lbrakk>retraction_map X Y r; locally_path_connected_space X\<rbrakk>
\<Longrightarrow> locally_path_connected_space Y"
using Abstract_Topology.retraction_imp_quotient_map locally_path_connected_space_quotient_map_image by blast
lemma locally_path_connected_space_euclideanreal: "locally_path_connected_space euclideanreal"
unfolding locally_path_connected_space_def neighbourhood_base_of
proof clarsimp
fix W and x :: "real"
assume "open W" "x \<in> W"
then obtain e where "e > 0" and e: "\<And>x'. \<bar>x' - x\<bar> < e \<longrightarrow> x' \<in> W"
by (auto simp: open_real)
then show "\<exists>U. open U \<and> (\<exists>V. path_connected V \<and> x \<in> U \<and> U \<subseteq> V \<and> V \<subseteq> W)"
by (force intro!: convex_imp_path_connected exI [where x = "{x-e<..<x+e}"])
qed
lemma locally_path_connected_space_discrete_topology:
"locally_path_connected_space (discrete_topology U)"
using locally_path_connected_space_open_path_components by fastforce
lemma path_component_eq_connected_component_of:
assumes "locally_path_connected_space X"
shows "(path_component_of_set X x = connected_component_of_set X x)"
proof (cases "x \<in> topspace X")
case True
then show ?thesis
using connectedin_connected_component_of [of X x]
apply (clarsimp simp add: connectedin_def connected_space_clopen_in topspace_subtopology_subset cong: conj_cong)
apply (drule_tac x="path_component_of_set X x" in spec)
by (metis assms closedin_closed_subtopology closedin_connected_component_of closedin_path_component_of_locally_path_connected_space inf.absorb_iff2 inf.orderE openin_path_component_of_locally_path_connected_space openin_subtopology path_component_of_eq_empty path_component_subset_connected_component_of)
next
case False
then show ?thesis
using connected_component_of_eq_empty path_component_of_eq_empty by fastforce
qed
lemma path_components_eq_connected_components_of:
"locally_path_connected_space X \<Longrightarrow> (path_components_of X = connected_components_of X)"
by (simp add: path_components_of_def connected_components_of_def image_def path_component_eq_connected_component_of)
lemma path_connected_eq_connected_space:
"locally_path_connected_space X
\<Longrightarrow> path_connected_space X \<longleftrightarrow> connected_space X"
by (metis connected_components_of_subset_sing path_components_eq_connected_components_of path_components_of_subset_singleton)
lemma locally_path_connected_space_product_topology:
"locally_path_connected_space(product_topology X I) \<longleftrightarrow>
topspace(product_topology X I) = {} \<or>
finite {i. i \<in> I \<and> ~path_connected_space(X i)} \<and>
(\<forall>i \<in> I. locally_path_connected_space(X i))"
(is "?lhs \<longleftrightarrow> ?empty \<or> ?rhs")
proof (cases ?empty)
case True
then show ?thesis
by (simp add: locally_path_connected_space_def neighbourhood_base_of openin_closedin_eq)
next
case False
then obtain z where z: "z \<in> (\<Pi>\<^sub>E i\<in>I. topspace (X i))"
by auto
have ?rhs if L: ?lhs
proof -
obtain U C where U: "openin (product_topology X I) U"
and V: "path_connectedin (product_topology X I) C"
and "z \<in> U" "U \<subseteq> C" and Csub: "C \<subseteq> (\<Pi>\<^sub>E i\<in>I. topspace (X i))"
using L apply (clarsimp simp add: locally_path_connected_space_def neighbourhood_base_of)
by (metis openin_topspace topspace_product_topology z)
then obtain V where finV: "finite {i \<in> I. V i \<noteq> topspace (X i)}"
and XV: "\<And>i. i\<in>I \<Longrightarrow> openin (X i) (V i)" and "z \<in> Pi\<^sub>E I V" and subU: "Pi\<^sub>E I V \<subseteq> U"
by (force simp: openin_product_topology_alt)
show ?thesis
proof (intro conjI ballI)
have "path_connected_space (X i)" if "i \<in> I" "V i = topspace (X i)" for i
proof -
have pc: "path_connectedin (X i) ((\<lambda>x. x i) ` C)"
apply (rule path_connectedin_continuous_map_image [OF _ V])
by (simp add: continuous_map_product_projection \<open>i \<in> I\<close>)
moreover have "((\<lambda>x. x i) ` C) = topspace (X i)"
proof
show "(\<lambda>x. x i) ` C \<subseteq> topspace (X i)"
by (simp add: pc path_connectedin_subset_topspace)
have "V i \<subseteq> (\<lambda>x. x i) ` (\<Pi>\<^sub>E i\<in>I. V i)"
by (metis \<open>z \<in> Pi\<^sub>E I V\<close> empty_iff image_projection_PiE order_refl that(1))
also have "\<dots> \<subseteq> (\<lambda>x. x i) ` U"
using subU by blast
finally show "topspace (X i) \<subseteq> (\<lambda>x. x i) ` C"
using \<open>U \<subseteq> C\<close> that by blast
qed
ultimately show ?thesis
by (simp add: path_connectedin_topspace)
qed
then have "{i \<in> I. \<not> path_connected_space (X i)} \<subseteq> {i \<in> I. V i \<noteq> topspace (X i)}"
by blast
with finV show "finite {i \<in> I. \<not> path_connected_space (X i)}"
using finite_subset by blast
next
show "locally_path_connected_space (X i)" if "i \<in> I" for i
apply (rule locally_path_connected_space_quotient_map_image [OF _ L, where f = "\<lambda>x. x i"])
by (metis False Abstract_Topology.retraction_imp_quotient_map retraction_map_product_projection that)
qed
qed
moreover have ?lhs if R: ?rhs
proof (clarsimp simp add: locally_path_connected_space_def neighbourhood_base_of)
fix F z
assume "openin (product_topology X I) F" and "z \<in> F"
then obtain W where finW: "finite {i \<in> I. W i \<noteq> topspace (X i)}"
and opeW: "\<And>i. i \<in> I \<Longrightarrow> openin (X i) (W i)" and "z \<in> Pi\<^sub>E I W" "Pi\<^sub>E I W \<subseteq> F"
by (auto simp: openin_product_topology_alt)
have "\<forall>i \<in> I. \<exists>U C. openin (X i) U \<and> path_connectedin (X i) C \<and> z i \<in> U \<and> U \<subseteq> C \<and> C \<subseteq> W i \<and>
(W i = topspace (X i) \<and>
path_connected_space (X i) \<longrightarrow> U = topspace (X i) \<and> C = topspace (X i))"
(is "\<forall>i \<in> I. ?\<Phi> i")
proof
fix i assume "i \<in> I"
have "locally_path_connected_space (X i)"
by (simp add: R \<open>i \<in> I\<close>)
moreover have "openin (X i) (W i) " "z i \<in> W i"
using \<open>z \<in> Pi\<^sub>E I W\<close> opeW \<open>i \<in> I\<close> by auto
ultimately obtain U C where UC: "openin (X i) U" "path_connectedin (X i) C" "z i \<in> U" "U \<subseteq> C" "C \<subseteq> W i"
using \<open>i \<in> I\<close> by (force simp: locally_path_connected_space_def neighbourhood_base_of)
show "?\<Phi> i"
proof (cases "W i = topspace (X i) \<and> path_connected_space(X i)")
case True
then show ?thesis
using \<open>z i \<in> W i\<close> path_connectedin_topspace by blast
next
case False
then show ?thesis
by (meson UC)
qed
qed
then obtain U C where
*: "\<And>i. i \<in> I \<Longrightarrow> openin (X i) (U i) \<and> path_connectedin (X i) (C i) \<and> z i \<in> (U i) \<and> (U i) \<subseteq> (C i) \<and> (C i) \<subseteq> W i \<and>
(W i = topspace (X i) \<and> path_connected_space (X i)
\<longrightarrow> (U i) = topspace (X i) \<and> (C i) = topspace (X i))"
by metis
let ?A = "{i \<in> I. \<not> path_connected_space (X i)} \<union> {i \<in> I. W i \<noteq> topspace (X i)}"
have "{i \<in> I. U i \<noteq> topspace (X i)} \<subseteq> ?A"
by (clarsimp simp add: "*")
moreover have "finite ?A"
by (simp add: that finW)
ultimately have "finite {i \<in> I. U i \<noteq> topspace (X i)}"
using finite_subset by auto
then have "openin (product_topology X I) (Pi\<^sub>E I U)"
using * by (simp add: openin_PiE_gen)
then show "\<exists>U. openin (product_topology X I) U \<and>
(\<exists>V. path_connectedin (product_topology X I) V \<and> z \<in> U \<and> U \<subseteq> V \<and> V \<subseteq> F)"
apply (rule_tac x="PiE I U" in exI, simp)
apply (rule_tac x="PiE I C" in exI)
using \<open>z \<in> Pi\<^sub>E I W\<close> \<open>Pi\<^sub>E I W \<subseteq> F\<close> *
apply (simp add: path_connectedin_PiE subset_PiE PiE_iff PiE_mono dual_order.trans)
done
qed
ultimately show ?thesis
using False by blast
qed
lemma locally_path_connected_is_realinterval:
assumes "is_interval S"
shows "locally_path_connected_space(subtopology euclideanreal S)"
unfolding locally_path_connected_space_def
proof (clarsimp simp add: neighbourhood_base_of openin_subtopology_alt)
fix a U
assume "a \<in> S" and "a \<in> U" and "open U"
then obtain r where "r > 0" and r: "ball a r \<subseteq> U"
by (metis open_contains_ball_eq)
show "\<exists>W. open W \<and> (\<exists>V. path_connectedin (top_of_set S) V \<and> a \<in> W \<and> S \<inter> W \<subseteq> V \<and> V \<subseteq> S \<and> V \<subseteq> U)"
proof (intro exI conjI)
show "path_connectedin (top_of_set S) (S \<inter> ball a r)"
by (simp add: assms is_interval_Int is_interval_ball_real is_interval_path_connected path_connectedin_subtopology)
show "a \<in> ball a r"
by (simp add: \<open>0 < r\<close>)
qed (use \<open>0 < r\<close> r in auto)
qed
lemma locally_path_connected_real_interval:
"locally_path_connected_space (subtopology euclideanreal{a..b})"
"locally_path_connected_space (subtopology euclideanreal{a<..<b})"
using locally_path_connected_is_realinterval
by (auto simp add: is_interval_1)
lemma locally_path_connected_space_prod_topology:
"locally_path_connected_space (prod_topology X Y) \<longleftrightarrow>
topspace (prod_topology X Y) = {} \<or>
locally_path_connected_space X \<and> locally_path_connected_space Y" (is "?lhs=?rhs")
proof (cases "topspace(prod_topology X Y) = {}")
case True
then show ?thesis
by (metis equals0D locally_path_connected_space_def neighbourhood_base_of_def)
next
case False
then have ne: "topspace X \<noteq> {}" "topspace Y \<noteq> {}"
by simp_all
show ?thesis
proof
assume ?lhs then show ?rhs
by (metis ne locally_path_connected_space_retraction_map_image retraction_map_fst retraction_map_snd)
next
assume ?rhs
with False have X: "locally_path_connected_space X" and Y: "locally_path_connected_space Y"
by auto
show ?lhs
unfolding locally_path_connected_space_def neighbourhood_base_of
proof clarify
fix UV x y
assume UV: "openin (prod_topology X Y) UV" and "(x,y) \<in> UV"
obtain A B where W12: "openin X A \<and> openin Y B \<and> x \<in> A \<and> y \<in> B \<and> A \<times> B \<subseteq> UV"
using X Y by (metis UV \<open>(x,y) \<in> UV\<close> openin_prod_topology_alt)
then obtain C D K L
where "openin X C" "path_connectedin X K" "x \<in> C" "C \<subseteq> K" "K \<subseteq> A"
"openin Y D" "path_connectedin Y L" "y \<in> D" "D \<subseteq> L" "L \<subseteq> B"
by (metis X Y locally_path_connected_space)
with W12 \<open>openin X C\<close> \<open>openin Y D\<close>
show "\<exists>U V. openin (prod_topology X Y) U \<and> path_connectedin (prod_topology X Y) V \<and> (x, y) \<in> U \<and> U \<subseteq> V \<and> V \<subseteq> UV"
apply (rule_tac x="C \<times> D" in exI)
apply (rule_tac x="K \<times> L" in exI)
apply (auto simp: openin_prod_Times_iff path_connectedin_Times)
done
qed
qed
qed
lemma locally_path_connected_space_sum_topology:
"locally_path_connected_space(sum_topology X I) \<longleftrightarrow>
(\<forall>i \<in> I. locally_path_connected_space (X i))" (is "?lhs=?rhs")
proof
assume ?lhs then show ?rhs
by (smt (verit) homeomorphic_locally_path_connected_space locally_path_connected_space_open_subset topological_property_of_sum_component)
next
assume R: ?rhs
show ?lhs
proof (clarsimp simp add: locally_path_connected_space_def neighbourhood_base_of forall_openin_sum_topology imp_conjL)
fix W i x
assume ope: "\<forall>i\<in>I. openin (X i) (W i)"
and "i \<in> I" and "x \<in> W i"
then obtain U V where U: "openin (X i) U" and V: "path_connectedin (X i) V"
and "x \<in> U" "U \<subseteq> V" "V \<subseteq> W i"
by (metis R \<open>i \<in> I\<close> \<open>x \<in> W i\<close> locally_path_connected_space)
show "\<exists>U. openin (sum_topology X I) U \<and> (\<exists>V. path_connectedin (sum_topology X I) V \<and> (i, x) \<in> U \<and> U \<subseteq> V \<and> V \<subseteq> Sigma I W)"
proof (intro exI conjI)
show "openin (sum_topology X I) (Pair i ` U)"
by (meson U \<open>i \<in> I\<close> open_map_component_injection open_map_def)
show "path_connectedin (sum_topology X I) (Pair i ` V)"
by (meson V \<open>i \<in> I\<close> continuous_map_component_injection path_connectedin_continuous_map_image)
show "Pair i ` V \<subseteq> Sigma I W"
using \<open>V \<subseteq> W i\<close> \<open>i \<in> I\<close> by force
qed (use \<open>x \<in> U\<close> \<open>U \<subseteq> V\<close> in auto)
qed
qed
subsection\<open>Locally connected spaces\<close>
definition weakly_locally_connected_at
where "weakly_locally_connected_at x X \<equiv> neighbourhood_base_at x (connectedin X) X"
definition locally_connected_at
where "locally_connected_at x X \<equiv>
neighbourhood_base_at x (\<lambda>U. openin X U \<and> connectedin X U ) X"
definition locally_connected_space
where "locally_connected_space X \<equiv> neighbourhood_base_of (connectedin X) X"
lemma locally_connected_A: "(\<forall>U x. openin X U \<and> x \<in> U
\<longrightarrow> openin X (connected_component_of_set (subtopology X U) x))
\<Longrightarrow> neighbourhood_base_of (\<lambda>U. openin X U \<and> connectedin X U) X"
by (smt (verit, best) connected_component_of_refl connectedin_connected_component_of connectedin_subtopology mem_Collect_eq neighbourhood_base_of openin_subset topspace_subtopology_subset)
lemma locally_connected_B: "locally_connected_space X \<Longrightarrow>
(\<forall>U x. openin X U \<and> x \<in> U \<longrightarrow> openin X (connected_component_of_set (subtopology X U) x))"
unfolding locally_connected_space_def neighbourhood_base_of
apply (erule all_forward)
apply clarify
apply (subst openin_subopen)
by (smt (verit, ccfv_threshold) Ball_Collect connected_component_of_def connected_component_of_equiv connectedin_subtopology in_mono mem_Collect_eq)
lemma locally_connected_C: "neighbourhood_base_of (\<lambda>U. openin X U \<and> connectedin X U) X \<Longrightarrow> locally_connected_space X"
using locally_connected_space_def neighbourhood_base_of_mono by auto
lemma locally_connected_space_alt:
"locally_connected_space X \<longleftrightarrow> neighbourhood_base_of (\<lambda>U. openin X U \<and> connectedin X U) X"
using locally_connected_A locally_connected_B locally_connected_C by blast
lemma locally_connected_space_eq_open_connected_component_of:
"locally_connected_space X \<longleftrightarrow>
(\<forall>U x. openin X U \<and> x \<in> U
\<longrightarrow> openin X (connected_component_of_set (subtopology X U) x))"
by (meson locally_connected_A locally_connected_B locally_connected_C)
lemma locally_connected_space:
"locally_connected_space X \<longleftrightarrow>
(\<forall>V x. openin X V \<and> x \<in> V \<longrightarrow> (\<exists>U. openin X U \<and> connectedin X U \<and> x \<in> U \<and> U \<subseteq> V))"
by (simp add: locally_connected_space_alt open_neighbourhood_base_of)
lemma locally_path_connected_imp_locally_connected_space:
"locally_path_connected_space X \<Longrightarrow> locally_connected_space X"
by (simp add: locally_connected_space_def locally_path_connected_space_def neighbourhood_base_of_mono path_connectedin_imp_connectedin)
lemma locally_connected_space_open_connected_components:
"locally_connected_space X \<longleftrightarrow>
(\<forall>U C. openin X U \<and> C \<in> connected_components_of(subtopology X U) \<longrightarrow> openin X C)"
apply (simp add: locally_connected_space_eq_open_connected_component_of connected_components_of_def)
by (smt (verit) imageE image_eqI inf.orderE inf_commute openin_subset)
lemma openin_connected_component_of_locally_connected_space:
"locally_connected_space X \<Longrightarrow> openin X (connected_component_of_set X x)"
by (metis connected_component_of_eq_empty locally_connected_space_eq_open_connected_component_of openin_empty openin_topspace subtopology_topspace)
lemma openin_connected_components_of_locally_connected_space:
"\<lbrakk>locally_connected_space X; C \<in> connected_components_of X\<rbrakk> \<Longrightarrow> openin X C"
by (metis locally_connected_space_open_connected_components openin_topspace subtopology_topspace)
lemma weakly_locally_connected_at:
"weakly_locally_connected_at x X \<longleftrightarrow>
(\<forall>V. openin X V \<and> x \<in> V
\<longrightarrow> (\<exists>U. openin X U \<and> x \<in> U \<and> U \<subseteq> V \<and>
(\<forall>y \<in> U. \<exists>C. connectedin X C \<and> C \<subseteq> V \<and> x \<in> C \<and> y \<in> C)))" (is "?lhs=?rhs")
proof
assume ?lhs then show ?rhs
unfolding neighbourhood_base_at_def weakly_locally_connected_at_def
by (meson subsetD subset_trans)
next
assume R: ?rhs
show ?lhs
unfolding neighbourhood_base_at_def weakly_locally_connected_at_def
proof clarify
fix V
assume "openin X V" and "x \<in> V"
then obtain U where "openin X U" "x \<in> U" "U \<subseteq> V"
and U: "\<forall>y\<in>U. \<exists>C. connectedin X C \<and> C \<subseteq> V \<and> x \<in> C \<and> y \<in> C"
using R by force
show "\<exists>A B. openin X A \<and> connectedin X B \<and> x \<in> A \<and> A \<subseteq> B \<and> B \<subseteq> V"
proof (intro conjI exI)
show "connectedin X (connected_component_of_set (subtopology X V) x)"
by (meson connectedin_connected_component_of connectedin_subtopology)
show "U \<subseteq> connected_component_of_set (subtopology X V) x"
using connected_component_of_maximal U
by (simp add: connected_component_of_def connectedin_subtopology subsetI)
show "connected_component_of_set (subtopology X V) x \<subseteq> V"
using connected_component_of_subset_topspace by fastforce
qed (auto simp: \<open>x \<in> U\<close> \<open>openin X U\<close>)
qed
qed
lemma locally_connected_space_iff_weak:
"locally_connected_space X \<longleftrightarrow> (\<forall>x \<in> topspace X. weakly_locally_connected_at x X)"
by (simp add: locally_connected_space_def neighbourhood_base_of_def weakly_locally_connected_at_def)
lemma locally_connected_space_im_kleinen:
"locally_connected_space X \<longleftrightarrow>
(\<forall>V x. openin X V \<and> x \<in> V
\<longrightarrow> (\<exists>U. openin X U \<and> x \<in> U \<and> U \<subseteq> V \<and>
(\<forall>y \<in> U. \<exists>C. connectedin X C \<and> C \<subseteq> V \<and> x \<in> C \<and> y \<in> C)))"
unfolding locally_connected_space_iff_weak weakly_locally_connected_at
using openin_subset subsetD by fastforce
lemma locally_connected_space_open_subset:
"\<lbrakk>locally_connected_space X; openin X S\<rbrakk> \<Longrightarrow> locally_connected_space (subtopology X S)"
apply (simp add: locally_connected_space_def)
by (smt (verit, ccfv_threshold) connectedin_subtopology neighbourhood_base_of openin_open_subtopology subset_trans)
lemma locally_connected_space_quotient_map_image:
assumes X: "locally_connected_space X" and f: "quotient_map X Y f"
shows "locally_connected_space Y"
unfolding locally_connected_space_open_connected_components
proof clarify
fix V C
assume "openin Y V" and C: "C \<in> connected_components_of (subtopology Y V)"
then have "C \<subseteq> topspace Y"
using connected_components_of_subset by force
have ope1: "openin X {a \<in> topspace X. f a \<in> V}"
using \<open>openin Y V\<close> f openin_continuous_map_preimage quotient_imp_continuous_map by blast
define Vf where "Vf \<equiv> {z \<in> topspace X. f z \<in> V}"
have "openin X {x \<in> topspace X. f x \<in> C}"
proof (clarsimp simp: openin_subopen [where S = "{x \<in> topspace X. f x \<in> C}"])
fix x
assume "x \<in> topspace X" and "f x \<in> C"
show "\<exists>T. openin X T \<and> x \<in> T \<and> T \<subseteq> {x \<in> topspace X. f x \<in> C}"
proof (intro exI conjI)
show "openin X (connected_component_of_set (subtopology X Vf) x)"
by (metis Vf_def X connected_component_of_eq_empty locally_connected_B ope1 openin_empty
openin_subset topspace_subtopology_subset)
show x_in_conn: "x \<in> connected_component_of_set (subtopology X Vf) x"
using C Vf_def \<open>f x \<in> C\<close> \<open>x \<in> topspace X\<close> connected_component_of_refl connected_components_of_subset by fastforce
have "connected_component_of_set (subtopology X Vf) x \<subseteq> topspace X \<inter> Vf"
using connected_component_of_subset_topspace by fastforce
moreover
have "f ` connected_component_of_set (subtopology X Vf) x \<subseteq> C"
proof (rule connected_components_of_maximal [where X = "subtopology Y V"])
show "C \<in> connected_components_of (subtopology Y V)"
by (simp add: C)
have \<section>: "quotient_map (subtopology X Vf) (subtopology Y V) f"
by (simp add: Vf_def \<open>openin Y V\<close> f quotient_map_restriction)
then show "connectedin (subtopology Y V) (f ` connected_component_of_set (subtopology X Vf) x)"
by (metis connectedin_connected_component_of connectedin_continuous_map_image quotient_imp_continuous_map)
show "\<not> disjnt C (f ` connected_component_of_set (subtopology X Vf) x)"
using \<open>f x \<in> C\<close> x_in_conn by (auto simp: disjnt_iff)
qed
ultimately
show "connected_component_of_set (subtopology X Vf) x \<subseteq> {x \<in> topspace X. f x \<in> C}"
by blast
qed
qed
then show "openin Y C"
using \<open>C \<subseteq> topspace Y\<close> f quotient_map_def by fastforce
qed
lemma locally_connected_space_retraction_map_image:
"\<lbrakk>retraction_map X Y r; locally_connected_space X\<rbrakk>
\<Longrightarrow> locally_connected_space Y"
using locally_connected_space_quotient_map_image retraction_imp_quotient_map by blast
lemma homeomorphic_locally_connected_space:
"X homeomorphic_space Y \<Longrightarrow> locally_connected_space X \<longleftrightarrow> locally_connected_space Y"
by (meson homeomorphic_map_def homeomorphic_space homeomorphic_space_sym locally_connected_space_quotient_map_image)
lemma locally_connected_space_euclideanreal: "locally_connected_space euclideanreal"
by (simp add: locally_path_connected_imp_locally_connected_space locally_path_connected_space_euclideanreal)
lemma locally_connected_is_realinterval:
"is_interval S \<Longrightarrow> locally_connected_space(subtopology euclideanreal S)"
by (simp add: locally_path_connected_imp_locally_connected_space locally_path_connected_is_realinterval)
lemma locally_connected_real_interval:
"locally_connected_space (subtopology euclideanreal{a..b})"
"locally_connected_space (subtopology euclideanreal{a<..<b})"
using connected_Icc is_interval_connected_1 locally_connected_is_realinterval by auto
lemma locally_connected_space_discrete_topology:
"locally_connected_space (discrete_topology U)"
by (simp add: locally_path_connected_imp_locally_connected_space locally_path_connected_space_discrete_topology)
lemma locally_path_connected_imp_locally_connected_at:
"locally_path_connected_at x X \<Longrightarrow> locally_connected_at x X"
by (simp add: locally_connected_at_def locally_path_connected_at_def neighbourhood_base_at_mono path_connectedin_imp_connectedin)
lemma weakly_locally_path_connected_imp_weakly_locally_connected_at:
"weakly_locally_path_connected_at x X
\<Longrightarrow> weakly_locally_connected_at x X"
by (simp add: neighbourhood_base_at_mono path_connectedin_imp_connectedin weakly_locally_connected_at_def weakly_locally_path_connected_at_def)
lemma interior_of_locally_connected_subspace_component:
assumes X: "locally_connected_space X"
and C: "C \<in> connected_components_of (subtopology X S)"
shows "X interior_of C = C \<inter> X interior_of S"
proof -
obtain Csub: "C \<subseteq> topspace X" "C \<subseteq> S"
by (meson C connectedin_connected_components_of connectedin_subset_topspace connectedin_subtopology)
show ?thesis
proof
show "X interior_of C \<subseteq> C \<inter> X interior_of S"
by (simp add: Csub interior_of_mono interior_of_subset)
have eq: "X interior_of S = \<Union> (connected_components_of (subtopology X (X interior_of S)))"
by (metis Union_connected_components_of interior_of_subset_topspace topspace_subtopology_subset)
moreover have "C \<inter> D \<subseteq> X interior_of C"
if "D \<in> connected_components_of (subtopology X (X interior_of S))" for D
proof (cases "C \<inter> D = {}")
case False
have "D \<subseteq> X interior_of C"
proof (rule interior_of_maximal)
have "connectedin (subtopology X S) D"
by (meson connectedin_connected_components_of connectedin_subtopology interior_of_subset subset_trans that)
then show "D \<subseteq> C"
by (meson C False connected_components_of_maximal disjnt_def)
show "openin X D"
using X locally_connected_space_open_connected_components openin_interior_of that by blast
qed
then show ?thesis
by blast
qed auto
ultimately show "C \<inter> X interior_of S \<subseteq> X interior_of C"
by blast
qed
qed
lemma frontier_of_locally_connected_subspace_component:
assumes X: "locally_connected_space X" and "closedin X S"
and C: "C \<in> connected_components_of (subtopology X S)"
shows "X frontier_of C = C \<inter> X frontier_of S"
proof -
obtain Csub: "C \<subseteq> topspace X" "C \<subseteq> S"
by (meson C connectedin_connected_components_of connectedin_subset_topspace connectedin_subtopology)
then have "X closure_of C - X interior_of C = C \<inter> X closure_of S - C \<inter> X interior_of S"
using assms
apply (simp add: closure_of_closedin flip: interior_of_locally_connected_subspace_component)
by (metis closedin_connected_components_of closedin_trans_full closure_of_eq inf.orderE)
then show ?thesis
by (simp add: Diff_Int_distrib frontier_of_def)
qed
(*Similar proof to locally_connected_space_prod_topology*)
lemma locally_connected_space_prod_topology:
"locally_connected_space (prod_topology X Y) \<longleftrightarrow>
topspace (prod_topology X Y) = {} \<or>
locally_connected_space X \<and> locally_connected_space Y" (is "?lhs=?rhs")
proof (cases "topspace(prod_topology X Y) = {}")
case True
then show ?thesis
using locally_connected_space_iff_weak by force
next
case False
then have ne: "topspace X \<noteq> {}" "topspace Y \<noteq> {}"
by simp_all
show ?thesis
proof
assume ?lhs then show ?rhs
by (metis locally_connected_space_quotient_map_image ne quotient_map_fst quotient_map_snd)
next
assume ?rhs
with False have X: "locally_connected_space X" and Y: "locally_connected_space Y"
by auto
show ?lhs
unfolding locally_connected_space_def neighbourhood_base_of
proof clarify
fix UV x y
assume UV: "openin (prod_topology X Y) UV" and "(x,y) \<in> UV"
obtain A B where W12: "openin X A \<and> openin Y B \<and> x \<in> A \<and> y \<in> B \<and> A \<times> B \<subseteq> UV"
using X Y by (metis UV \<open>(x,y) \<in> UV\<close> openin_prod_topology_alt)
then obtain C D K L
where "openin X C" "connectedin X K" "x \<in> C" "C \<subseteq> K" "K \<subseteq> A"
"openin Y D" "connectedin Y L" "y \<in> D" "D \<subseteq> L" "L \<subseteq> B"
by (metis X Y locally_connected_space)
with W12 \<open>openin X C\<close> \<open>openin Y D\<close>
show "\<exists>U V. openin (prod_topology X Y) U \<and> connectedin (prod_topology X Y) V \<and> (x, y) \<in> U \<and> U \<subseteq> V \<and> V \<subseteq> UV"
apply (rule_tac x="C \<times> D" in exI)
apply (rule_tac x="K \<times> L" in exI)
apply (auto simp: openin_prod_Times_iff connectedin_Times)
done
qed
qed
qed
(*Same proof as locally_path_connected_space_product_topology*)
lemma locally_connected_space_product_topology:
"locally_connected_space(product_topology X I) \<longleftrightarrow>
topspace(product_topology X I) = {} \<or>
finite {i. i \<in> I \<and> ~connected_space(X i)} \<and>
(\<forall>i \<in> I. locally_connected_space(X i))"
(is "?lhs \<longleftrightarrow> ?empty \<or> ?rhs")
proof (cases ?empty)
case True
then show ?thesis
by (simp add: locally_connected_space_def neighbourhood_base_of openin_closedin_eq)
next
case False
then obtain z where z: "z \<in> (\<Pi>\<^sub>E i\<in>I. topspace (X i))"
by auto
have ?rhs if L: ?lhs
proof -
obtain U C where U: "openin (product_topology X I) U"
and V: "connectedin (product_topology X I) C"
and "z \<in> U" "U \<subseteq> C" and Csub: "C \<subseteq> (\<Pi>\<^sub>E i\<in>I. topspace (X i))"
using L apply (clarsimp simp add: locally_connected_space_def neighbourhood_base_of)
by (metis openin_topspace topspace_product_topology z)
then obtain V where finV: "finite {i \<in> I. V i \<noteq> topspace (X i)}"
and XV: "\<And>i. i\<in>I \<Longrightarrow> openin (X i) (V i)" and "z \<in> Pi\<^sub>E I V" and subU: "Pi\<^sub>E I V \<subseteq> U"
by (force simp: openin_product_topology_alt)
show ?thesis
proof (intro conjI ballI)
have "connected_space (X i)" if "i \<in> I" "V i = topspace (X i)" for i
proof -
have pc: "connectedin (X i) ((\<lambda>x. x i) ` C)"
apply (rule connectedin_continuous_map_image [OF _ V])
by (simp add: continuous_map_product_projection \<open>i \<in> I\<close>)
moreover have "((\<lambda>x. x i) ` C) = topspace (X i)"
proof
show "(\<lambda>x. x i) ` C \<subseteq> topspace (X i)"
by (simp add: pc connectedin_subset_topspace)
have "V i \<subseteq> (\<lambda>x. x i) ` (\<Pi>\<^sub>E i\<in>I. V i)"
by (metis \<open>z \<in> Pi\<^sub>E I V\<close> empty_iff image_projection_PiE order_refl that(1))
also have "\<dots> \<subseteq> (\<lambda>x. x i) ` U"
using subU by blast
finally show "topspace (X i) \<subseteq> (\<lambda>x. x i) ` C"
using \<open>U \<subseteq> C\<close> that by blast
qed
ultimately show ?thesis
by (simp add: connectedin_topspace)
qed
then have "{i \<in> I. \<not> connected_space (X i)} \<subseteq> {i \<in> I. V i \<noteq> topspace (X i)}"
by blast
with finV show "finite {i \<in> I. \<not> connected_space (X i)}"
using finite_subset by blast
next
show "locally_connected_space (X i)" if "i \<in> I" for i
by (meson False L locally_connected_space_quotient_map_image quotient_map_product_projection that)
qed
qed
moreover have ?lhs if R: ?rhs
proof (clarsimp simp add: locally_connected_space_def neighbourhood_base_of)
fix F z
assume "openin (product_topology X I) F" and "z \<in> F"
then obtain W where finW: "finite {i \<in> I. W i \<noteq> topspace (X i)}"
and opeW: "\<And>i. i \<in> I \<Longrightarrow> openin (X i) (W i)" and "z \<in> Pi\<^sub>E I W" "Pi\<^sub>E I W \<subseteq> F"
by (auto simp: openin_product_topology_alt)
have "\<forall>i \<in> I. \<exists>U C. openin (X i) U \<and> connectedin (X i) C \<and> z i \<in> U \<and> U \<subseteq> C \<and> C \<subseteq> W i \<and>
(W i = topspace (X i) \<and>
connected_space (X i) \<longrightarrow> U = topspace (X i) \<and> C = topspace (X i))"
(is "\<forall>i \<in> I. ?\<Phi> i")
proof
fix i assume "i \<in> I"
have "locally_connected_space (X i)"
by (simp add: R \<open>i \<in> I\<close>)
moreover have "openin (X i) (W i) " "z i \<in> W i"
using \<open>z \<in> Pi\<^sub>E I W\<close> opeW \<open>i \<in> I\<close> by auto
ultimately obtain U C where UC: "openin (X i) U" "connectedin (X i) C" "z i \<in> U" "U \<subseteq> C" "C \<subseteq> W i"
using \<open>i \<in> I\<close> by (force simp: locally_connected_space_def neighbourhood_base_of)
show "?\<Phi> i"
proof (cases "W i = topspace (X i) \<and> connected_space(X i)")
case True
then show ?thesis
using \<open>z i \<in> W i\<close> connectedin_topspace by blast
next
case False
then show ?thesis
by (meson UC)
qed
qed
then obtain U C where
*: "\<And>i. i \<in> I \<Longrightarrow> openin (X i) (U i) \<and> connectedin (X i) (C i) \<and> z i \<in> (U i) \<and> (U i) \<subseteq> (C i) \<and> (C i) \<subseteq> W i \<and>
(W i = topspace (X i) \<and> connected_space (X i)
\<longrightarrow> (U i) = topspace (X i) \<and> (C i) = topspace (X i))"
by metis
let ?A = "{i \<in> I. \<not> connected_space (X i)} \<union> {i \<in> I. W i \<noteq> topspace (X i)}"
have "{i \<in> I. U i \<noteq> topspace (X i)} \<subseteq> ?A"
by (clarsimp simp add: "*")
moreover have "finite ?A"
by (simp add: that finW)
ultimately have "finite {i \<in> I. U i \<noteq> topspace (X i)}"
using finite_subset by auto
then have "openin (product_topology X I) (Pi\<^sub>E I U)"
using * by (simp add: openin_PiE_gen)
then show "\<exists>U. openin (product_topology X I) U \<and>
(\<exists>V. connectedin (product_topology X I) V \<and> z \<in> U \<and> U \<subseteq> V \<and> V \<subseteq> F)"
apply (rule_tac x="PiE I U" in exI, simp)
apply (rule_tac x="PiE I C" in exI)
using \<open>z \<in> Pi\<^sub>E I W\<close> \<open>Pi\<^sub>E I W \<subseteq> F\<close> *
apply (simp add: connectedin_PiE subset_PiE PiE_iff PiE_mono dual_order.trans)
done
qed
ultimately show ?thesis
using False by blast
qed
lemma locally_connected_space_sum_topology:
"locally_connected_space(sum_topology X I) \<longleftrightarrow>
(\<forall>i \<in> I. locally_connected_space (X i))" (is "?lhs=?rhs")
proof
assume ?lhs then show ?rhs
by (smt (verit) homeomorphic_locally_connected_space locally_connected_space_open_subset topological_property_of_sum_component)
next
assume R: ?rhs
show ?lhs
proof (clarsimp simp add: locally_connected_space_def neighbourhood_base_of forall_openin_sum_topology imp_conjL)
fix W i x
assume ope: "\<forall>i\<in>I. openin (X i) (W i)"
and "i \<in> I" and "x \<in> W i"
then obtain U V where U: "openin (X i) U" and V: "connectedin (X i) V"
and "x \<in> U" "U \<subseteq> V" "V \<subseteq> W i"
by (metis R \<open>i \<in> I\<close> \<open>x \<in> W i\<close> locally_connected_space)
show "\<exists>U. openin (sum_topology X I) U \<and> (\<exists>V. connectedin (sum_topology X I) V \<and> (i,x) \<in> U \<and> U \<subseteq> V \<and> V \<subseteq> Sigma I W)"
proof (intro exI conjI)
show "openin (sum_topology X I) (Pair i ` U)"
by (meson U \<open>i \<in> I\<close> open_map_component_injection open_map_def)
show "connectedin (sum_topology X I) (Pair i ` V)"
by (meson V \<open>i \<in> I\<close> continuous_map_component_injection connectedin_continuous_map_image)
show "Pair i ` V \<subseteq> Sigma I W"
using \<open>V \<subseteq> W i\<close> \<open>i \<in> I\<close> by force
qed (use \<open>x \<in> U\<close> \<open>U \<subseteq> V\<close> in auto)
qed
qed
subsection \<open>Dimension of a topological space\<close>
text\<open>Basic definition of the small inductive dimension relation. Works in any topological space.\<close>
inductive dimension_le :: "['a topology, int] \<Rightarrow> bool" (infix "dim'_le" 50)
where "\<lbrakk>-1 \<le> n;
\<And>V a. \<lbrakk>openin X V; a \<in> V\<rbrakk> \<Longrightarrow> \<exists>U. a \<in> U \<and> U \<subseteq> V \<and> openin X U \<and> (subtopology X (X frontier_of U)) dim_le (n-1)\<rbrakk>
\<Longrightarrow> X dim_le (n::int)"
lemma dimension_le_neighbourhood_base:
"X dim_le n \<longleftrightarrow>
-1 \<le> n \<and> neighbourhood_base_of (\<lambda>U. openin X U \<and> (subtopology X (X frontier_of U)) dim_le (n-1)) X"
by (smt (verit, best) dimension_le.simps open_neighbourhood_base_of)
lemma dimension_le_bound: "X dim_le n \<Longrightarrow>-1 \<le> n"
using dimension_le.simps by blast
lemma dimension_le_mono [rule_format]:
assumes "X dim_le m"
shows "m \<le> n \<longrightarrow> X dim_le n"
using assms
proof (induction arbitrary: n rule: dimension_le.induct)
case (1 m X)
show ?case
proof (intro strip dimension_le.intros)
show "-1 \<le> n" if "m \<le> n" for n :: int using that using "1.hyps" by fastforce
show "\<exists>U. a \<in> U \<and> U \<subseteq> V \<and> openin X U \<and> subtopology X (X frontier_of U) dim_le n-1"
if "m \<le> n" and "openin X V" and "a \<in> V" for n V a
using that by (meson "1.IH" diff_right_mono)
qed
qed
lemma dimension_le_eq_empty:
"X dim_le -1 \<longleftrightarrow> topspace X = {}"
proof
assume "X dim_le (-1)"
then show "topspace X = {}"
by (smt (verit, ccfv_threshold) Diff_empty Diff_eq_empty_iff dimension_le.cases openin_topspace subset_eq)
next
assume "topspace X = {}"
then show "X dim_le (-1)"
using dimension_le.simps openin_subset by fastforce
qed
lemma dimension_le_0_neighbourhood_base_of_clopen:
"X dim_le 0 \<longleftrightarrow> neighbourhood_base_of (\<lambda>U. closedin X U \<and> openin X U) X"
proof -
have "(subtopology X (X frontier_of U) dim_le -1) =
closedin X U" if "openin X U" for U
by (metis dimension_le_eq_empty frontier_of_eq_empty frontier_of_subset_topspace openin_subset that topspace_subtopology_subset)
then show ?thesis
by (smt (verit, del_insts) dimension_le.simps open_neighbourhood_base_of)
qed
lemma dimension_le_subtopology:
"X dim_le n \<Longrightarrow> subtopology X S dim_le n"
proof (induction arbitrary: S rule: dimension_le.induct)
case (1 n X)
show ?case
proof (intro dimension_le.intros)
show "- 1 \<le> n"
by (simp add: "1.hyps")
fix U' a
assume U': "openin (subtopology X S) U'" and "a \<in> U'"
then obtain U where U: "openin X U" "U' = U \<inter> S"
by (meson openin_subtopology)
then obtain V where "a \<in> V" "V \<subseteq> U" "openin X V"
and subV: "subtopology X (X frontier_of V) dim_le n-1"
and dimV: "\<And>T. subtopology X (X frontier_of V \<inter> T) dim_le n-1"
by (metis "1.IH" Int_iff \<open>a \<in> U'\<close> subtopology_subtopology)
show "\<exists>W. a \<in> W \<and> W \<subseteq> U' \<and> openin (subtopology X S) W \<and> subtopology (subtopology X S) (subtopology X S frontier_of W) dim_le n-1"
proof (intro exI conjI)
show "a \<in> S \<inter> V" "S \<inter> V \<subseteq> U'"
using \<open>U' = U \<inter> S\<close> \<open>a \<in> U'\<close> \<open>a \<in> V\<close> \<open>V \<subseteq> U\<close> by blast+
show "openin (subtopology X S) (S \<inter> V)"
by (simp add: \<open>openin X V\<close> openin_subtopology_Int2)
have "S \<inter> subtopology X S frontier_of V \<subseteq> X frontier_of V"
by (simp add: frontier_of_subtopology_subset)
then show "subtopology (subtopology X S) (subtopology X S frontier_of (S \<inter> V)) dim_le n-1"
by (metis dimV frontier_of_restrict inf.absorb_iff2 inf_left_idem subtopology_subtopology topspace_subtopology)
qed
qed
qed
lemma dimension_le_subtopologies:
"\<lbrakk>subtopology X T dim_le n; S \<subseteq> T\<rbrakk> \<Longrightarrow> (subtopology X S) dim_le n"
by (metis dimension_le_subtopology inf.absorb_iff2 subtopology_subtopology)
lemma dimension_le_eq_subtopology:
"(subtopology X S) dim_le n \<longleftrightarrow>
-1 \<le> n \<and>
(\<forall>V a. openin X V \<and> a \<in> V \<and> a \<in> S
\<longrightarrow> (\<exists>U. a \<in> U \<and> U \<subseteq> V \<and> openin X U \<and>
subtopology X (subtopology X S frontier_of (S \<inter> U)) dim_le (n-1)))"
proof -
have *: "(\<exists>T. a \<in> T \<and> T \<inter> S \<subseteq> V \<inter> S \<and> openin X T \<and> subtopology X (S \<inter> (subtopology X S frontier_of (T \<inter> S))) dim_le n-1)
\<longleftrightarrow> (\<exists>U. a \<in> U \<and> U \<subseteq> V \<and> openin X U \<and> subtopology X (subtopology X S frontier_of (S \<inter> U)) dim_le n-1)"
if "a \<in> V" "a \<in> S" "openin X V" for a V
proof -
have "\<exists>U. a \<in> U \<and> U \<subseteq> V \<and> openin X U \<and> subtopology X (subtopology X S frontier_of (S \<inter> U)) dim_le n-1"
if "a \<in> T" and sub: "T \<inter> S \<subseteq> V \<inter> S" and "openin X T"
and dim: "subtopology X (S \<inter> subtopology X S frontier_of (T \<inter> S)) dim_le n-1"
for T
proof (intro exI conjI)
show "openin X (T \<inter> V)"
using \<open>openin X V\<close> \<open>openin X T\<close> by blast
show "subtopology X (subtopology X S frontier_of (S \<inter> (T \<inter> V))) dim_le n-1"
by (metis dim frontier_of_subset_subtopology inf.boundedE inf_absorb2 inf_assoc inf_commute sub)
qed (use \<open>a \<in> V\<close> \<open>a \<in> T\<close> in auto)
moreover have "\<exists>T. a \<in> T \<and> T \<inter> S \<subseteq> V \<inter> S \<and> openin X T \<and> subtopology X (S \<inter> subtopology X S frontier_of (T \<inter> S)) dim_le n-1"
if "a \<in> U" and "U \<subseteq> V" and "openin X U"
and dim: "subtopology X (subtopology X S frontier_of (S \<inter> U)) dim_le n-1"
for U
by (metis that frontier_of_subset_subtopology inf_absorb2 inf_commute inf_le1 le_inf_iff)
ultimately show ?thesis
by safe
qed
show ?thesis
apply (simp add: dimension_le.simps [of _ n] subtopology_subtopology openin_subtopology flip: *)
by (safe; metis Int_iff inf_le2 le_inf_iff)
qed
lemma homeomorphic_space_dimension_le_aux:
assumes "X homeomorphic_space Y" "X dim_le of_nat n - 1"
shows "Y dim_le of_nat n - 1"
using assms
proof (induction n arbitrary: X Y)
case 0
then show ?case
by (simp add: dimension_le_eq_empty homeomorphic_empty_space)
next
case (Suc n)
then have X_dim_n: "X dim_le n"
by simp
show ?case
proof (clarsimp simp add: dimension_le.simps [of Y n])
fix V b
assume "openin Y V" and "b \<in> V"
obtain f g where fg: "homeomorphic_maps X Y f g"
using \<open>X homeomorphic_space Y\<close> homeomorphic_space_def by blast
then have "openin X (g ` V)"
using \<open>openin Y V\<close> homeomorphic_map_openness_eq homeomorphic_maps_map by blast
then obtain U where "g b \<in> U" "openin X U" and gim: "U \<subseteq> g ` V" and sub: "subtopology X (X frontier_of U) dim_le int n - int 1"
using X_dim_n unfolding dimension_le.simps [of X n] by (metis \<open>b \<in> V\<close> imageI of_nat_eq_1_iff)
show "\<exists>U. b \<in> U \<and> U \<subseteq> V \<and> openin Y U \<and> subtopology Y (Y frontier_of U) dim_le int n - 1"
proof (intro conjI exI)
show "b \<in> f ` U"
by (metis (no_types, lifting) \<open>b \<in> V\<close> \<open>g b \<in> U\<close> \<open>openin Y V\<close> fg homeomorphic_maps_map image_iff openin_subset subsetD)
show "f ` U \<subseteq> V"
by (smt (verit, ccfv_threshold) \<open>openin Y V\<close> fg gim homeomorphic_maps_map image_iff openin_subset subset_iff)
show "openin Y (f ` U)"
using \<open>openin X U\<close> fg homeomorphic_map_openness_eq homeomorphic_maps_map by blast
show "subtopology Y (Y frontier_of f ` U) dim_le int n-1"
proof (rule Suc.IH)
have "homeomorphic_maps (subtopology X (X frontier_of U)) (subtopology Y (Y frontier_of f ` U)) f g"
using \<open>openin X U\<close> fg
by (metis frontier_of_subset_topspace homeomorphic_map_frontier_of homeomorphic_maps_map homeomorphic_maps_subtopologies openin_subset topspace_subtopology topspace_subtopology_subset)
then show "subtopology X (X frontier_of U) homeomorphic_space subtopology Y (Y frontier_of f ` U)"
using homeomorphic_space_def by blast
show "subtopology X (X frontier_of U) dim_le int n-1"
using sub by fastforce
qed
qed
qed
qed
lemma homeomorphic_space_dimension_le:
assumes "X homeomorphic_space Y"
shows "X dim_le n \<longleftrightarrow> Y dim_le n"
proof (cases "n \<ge> -1")
case True
then show ?thesis
using homeomorphic_space_dimension_le_aux [of _ _ "nat(n+1)"] by (smt (verit) assms homeomorphic_space_sym nat_eq_iff)
next
case False
then show ?thesis
by (metis dimension_le_bound)
qed
lemma dimension_le_retraction_map_image:
"\<lbrakk>retraction_map X Y r; X dim_le n\<rbrakk> \<Longrightarrow> Y dim_le n"
by (meson dimension_le_subtopology homeomorphic_space_dimension_le retraction_map_def retraction_maps_section_image2)
lemma dimension_le_discrete_topology [simp]: "(discrete_topology U) dim_le 0"
using dimension_le.simps dimension_le_eq_empty by fastforce
end