author | paulson <lp15@cam.ac.uk> |
Mon, 15 May 2023 17:12:18 +0100 | |
changeset 78037 | 37894dff0111 |
parent 71172 | 575b3a818de5 |
child 78250 | 400aecdfd71f |
permissions | -rw-r--r-- |
71137 | 1 |
section \<open>Neighbourhood bases and Locally path-connected spaces\<close> |
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theory Locally |
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imports |
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78037
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
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Path_Connected Function_Topology Sum_Topology |
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begin |
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subsection\<open>Neighbourhood Bases\<close> |
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text \<open>Useful for "local" properties of various kinds\<close> |
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definition neighbourhood_base_at where |
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"neighbourhood_base_at x P X \<equiv> |
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\<forall>W. openin X W \<and> x \<in> W |
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\<longrightarrow> (\<exists>U V. openin X U \<and> P V \<and> x \<in> U \<and> U \<subseteq> V \<and> V \<subseteq> W)" |
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definition neighbourhood_base_of where |
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"neighbourhood_base_of P X \<equiv> |
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\<forall>x \<in> topspace X. neighbourhood_base_at x P X" |
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lemma neighbourhood_base_of: |
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"neighbourhood_base_of P X \<longleftrightarrow> |
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(\<forall>W x. openin X W \<and> x \<in> W |
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\<longrightarrow> (\<exists>U V. openin X U \<and> P V \<and> x \<in> U \<and> U \<subseteq> V \<and> V \<subseteq> W))" |
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unfolding neighbourhood_base_at_def neighbourhood_base_of_def |
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using openin_subset by blast |
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lemma neighbourhood_base_at_mono: |
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"\<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" |
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unfolding neighbourhood_base_at_def |
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by (meson subset_eq) |
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lemma neighbourhood_base_of_mono: |
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"\<lbrakk>neighbourhood_base_of P X; \<And>S. P S \<Longrightarrow> Q S\<rbrakk> \<Longrightarrow> neighbourhood_base_of Q X" |
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unfolding neighbourhood_base_of_def |
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using neighbourhood_base_at_mono by force |
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lemma open_neighbourhood_base_at: |
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"(\<And>S. \<lbrakk>P S; x \<in> S\<rbrakk> \<Longrightarrow> openin X S) |
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\<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))" |
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unfolding neighbourhood_base_at_def |
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by (meson subsetD) |
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lemma open_neighbourhood_base_of: |
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"(\<forall>S. P S \<longrightarrow> openin X S) |
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\<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))" |
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apply (simp add: neighbourhood_base_of, safe, blast) |
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by meson |
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lemma neighbourhood_base_of_open_subset: |
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"\<lbrakk>neighbourhood_base_of P X; openin X S\<rbrakk> |
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\<Longrightarrow> neighbourhood_base_of P (subtopology X S)" |
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apply (clarsimp simp add: neighbourhood_base_of openin_subtopology_alt image_def) |
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apply (rename_tac x V) |
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apply (drule_tac x="S \<inter> V" in spec) |
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apply (simp add: Int_ac) |
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by (metis IntI le_infI1 openin_Int) |
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lemma neighbourhood_base_of_topology_base: |
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"openin X = arbitrary union_of (\<lambda>W. W \<in> \<B>) |
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\<Longrightarrow> neighbourhood_base_of P X \<longleftrightarrow> |
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(\<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))" |
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apply (auto simp: openin_topology_base_unique neighbourhood_base_of) |
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by (meson subset_trans) |
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lemma neighbourhood_base_at_unlocalized: |
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assumes "\<And>S T. \<lbrakk>P S; openin X T; x \<in> T; T \<subseteq> S\<rbrakk> \<Longrightarrow> P T" |
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shows "neighbourhood_base_at x P X |
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\<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))" |
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(is "?lhs = ?rhs") |
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proof |
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assume R: ?rhs show ?lhs |
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unfolding neighbourhood_base_at_def |
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proof clarify |
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fix W |
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assume "openin X W" "x \<in> W" |
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then have "x \<in> topspace X" |
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using openin_subset by blast |
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with R obtain U V where "openin X U" "P V" "x \<in> U" "U \<subseteq> V" "V \<subseteq> topspace X" |
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by blast |
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then show "\<exists>U V. openin X U \<and> P V \<and> x \<in> U \<and> U \<subseteq> V \<and> V \<subseteq> W" |
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by (metis IntI \<open>openin X W\<close> \<open>x \<in> W\<close> assms inf_le1 inf_le2 openin_Int) |
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qed |
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qed (simp add: neighbourhood_base_at_def) |
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lemma neighbourhood_base_at_with_subset: |
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"\<lbrakk>openin X U; x \<in> U\<rbrakk> |
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\<Longrightarrow> (neighbourhood_base_at x P X \<longleftrightarrow> neighbourhood_base_at x (\<lambda>T. T \<subseteq> U \<and> P T) X)" |
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apply (simp add: neighbourhood_base_at_def) |
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apply (metis IntI Int_subset_iff openin_Int) |
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done |
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lemma neighbourhood_base_of_with_subset: |
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"neighbourhood_base_of P X \<longleftrightarrow> neighbourhood_base_of (\<lambda>t. t \<subseteq> topspace X \<and> P t) X" |
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using neighbourhood_base_at_with_subset |
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by (fastforce simp add: neighbourhood_base_of_def) |
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subsection\<open>Locally path-connected spaces\<close> |
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definition weakly_locally_path_connected_at |
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where "weakly_locally_path_connected_at x X \<equiv> neighbourhood_base_at x (path_connectedin X) X" |
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definition locally_path_connected_at |
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where "locally_path_connected_at x X \<equiv> |
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neighbourhood_base_at x (\<lambda>U. openin X U \<and> path_connectedin X U) X" |
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definition locally_path_connected_space |
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where "locally_path_connected_space X \<equiv> neighbourhood_base_of (path_connectedin X) X" |
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lemma locally_path_connected_space_alt: |
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"locally_path_connected_space X \<longleftrightarrow> neighbourhood_base_of (\<lambda>U. openin X U \<and> path_connectedin X U) X" |
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(is "?P = ?Q") |
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and locally_path_connected_space_eq_open_path_component_of: |
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"locally_path_connected_space X \<longleftrightarrow> |
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(\<forall>U x. openin X U \<and> x \<in> U \<longrightarrow> openin X (Collect (path_component_of (subtopology X U) x)))" |
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(is "?P = ?R") |
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proof - |
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have ?P if ?Q |
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using locally_path_connected_space_def neighbourhood_base_of_mono that by auto |
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moreover have ?R if P: ?P |
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proof - |
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show ?thesis |
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proof clarify |
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fix U y |
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assume "openin X U" "y \<in> U" |
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have "\<exists>T. openin X T \<and> x \<in> T \<and> T \<subseteq> Collect (path_component_of (subtopology X U) y)" |
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if "path_component_of (subtopology X U) y x" for x |
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proof - |
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have "x \<in> U" |
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using path_component_of_equiv that topspace_subtopology by fastforce |
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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)" |
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using P |
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by (simp add: \<open>openin X U\<close> locally_path_connected_space_def neighbourhood_base_of) |
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then show ?thesis |
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by (metis dual_order.trans path_component_of_equiv path_component_of_maximal path_connectedin_subtopology subset_iff that) |
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qed |
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then show "openin X (Collect (path_component_of (subtopology X U) y))" |
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using openin_subopen by force |
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qed |
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qed |
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moreover have ?Q if ?R |
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using that |
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apply (simp add: open_neighbourhood_base_of) |
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by (metis mem_Collect_eq openin_subset path_component_of_refl path_connectedin_path_component_of path_connectedin_subtopology that topspace_subtopology_subset) |
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ultimately show "?P = ?Q" "?P = ?R" |
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by blast+ |
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qed |
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lemma locally_path_connected_space: |
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"locally_path_connected_space X |
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\<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))" |
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by (simp add: locally_path_connected_space_alt open_neighbourhood_base_of) |
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lemma locally_path_connected_space_open_path_components: |
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"locally_path_connected_space X \<longleftrightarrow> |
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(\<forall>U c. openin X U \<and> c \<in> path_components_of(subtopology X U) \<longrightarrow> openin X c)" |
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apply (auto simp: locally_path_connected_space_eq_open_path_component_of path_components_of_def) |
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by (metis imageI inf.absorb_iff2 openin_closedin_eq) |
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lemma openin_path_component_of_locally_path_connected_space: |
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"locally_path_connected_space X \<Longrightarrow> openin X (Collect (path_component_of X x))" |
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apply (auto simp: locally_path_connected_space_eq_open_path_component_of) |
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by (metis openin_empty openin_topspace path_component_of_eq_empty subtopology_topspace) |
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lemma openin_path_components_of_locally_path_connected_space: |
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"\<lbrakk>locally_path_connected_space X; c \<in> path_components_of X\<rbrakk> \<Longrightarrow> openin X c" |
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apply (auto simp: locally_path_connected_space_eq_open_path_component_of) |
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by (metis (no_types, lifting) imageE openin_topspace path_components_of_def subtopology_topspace) |
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lemma closedin_path_components_of_locally_path_connected_space: |
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"\<lbrakk>locally_path_connected_space X; c \<in> path_components_of X\<rbrakk> \<Longrightarrow> closedin X c" |
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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) |
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lemma closedin_path_component_of_locally_path_connected_space: |
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assumes "locally_path_connected_space X" |
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shows "closedin X (Collect (path_component_of X x))" |
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proof (cases "x \<in> topspace X") |
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case True |
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then show ?thesis |
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by (simp add: assms closedin_path_components_of_locally_path_connected_space path_component_in_path_components_of) |
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next |
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case False |
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then show ?thesis |
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by (metis closedin_empty path_component_of_eq_empty) |
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qed |
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lemma weakly_locally_path_connected_at: |
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"weakly_locally_path_connected_at x X \<longleftrightarrow> |
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(\<forall>V. openin X V \<and> x \<in> V |
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\<longrightarrow> (\<exists>U. openin X U \<and> x \<in> U \<and> U \<subseteq> V \<and> |
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(\<forall>y \<in> U. \<exists>C. path_connectedin X C \<and> C \<subseteq> V \<and> x \<in> C \<and> y \<in> C)))" |
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(is "?lhs = ?rhs") |
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proof |
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assume ?lhs then show ?rhs |
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apply (simp add: weakly_locally_path_connected_at_def neighbourhood_base_at_def) |
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by (meson order_trans subsetD) |
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next |
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have *: "\<exists>V. path_connectedin X V \<and> U \<subseteq> V \<and> V \<subseteq> W" |
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if "(\<forall>y\<in>U. \<exists>C. path_connectedin X C \<and> C \<subseteq> W \<and> x \<in> C \<and> y \<in> C)" |
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for W U |
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proof (intro exI conjI) |
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let ?V = "(Collect (path_component_of (subtopology X W) x))" |
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show "path_connectedin X (Collect (path_component_of (subtopology X W) x))" |
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by (meson path_connectedin_path_component_of path_connectedin_subtopology) |
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show "U \<subseteq> ?V" |
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by (auto simp: path_component_of path_connectedin_subtopology that) |
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show "?V \<subseteq> W" |
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by (meson path_connectedin_path_component_of path_connectedin_subtopology) |
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qed |
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assume ?rhs |
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then show ?lhs |
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unfolding weakly_locally_path_connected_at_def neighbourhood_base_at_def by (metis "*") |
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qed |
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lemma locally_path_connected_space_im_kleinen: |
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"locally_path_connected_space X \<longleftrightarrow> |
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(\<forall>V x. openin X V \<and> x \<in> V |
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\<longrightarrow> (\<exists>U. openin X U \<and> |
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x \<in> U \<and> U \<subseteq> V \<and> |
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(\<forall>y \<in> U. \<exists>c. path_connectedin X c \<and> |
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c \<subseteq> V \<and> x \<in> c \<and> y \<in> c)))" |
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apply (simp add: locally_path_connected_space_def neighbourhood_base_of_def) |
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apply (simp add: weakly_locally_path_connected_at flip: weakly_locally_path_connected_at_def) |
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using openin_subset apply force |
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done |
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lemma locally_path_connected_space_open_subset: |
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"\<lbrakk>locally_path_connected_space X; openin X s\<rbrakk> |
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\<Longrightarrow> locally_path_connected_space (subtopology X s)" |
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apply (simp add: locally_path_connected_space_def neighbourhood_base_of openin_open_subtopology path_connectedin_subtopology) |
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by (meson order_trans) |
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lemma locally_path_connected_space_quotient_map_image: |
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assumes f: "quotient_map X Y f" and X: "locally_path_connected_space X" |
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shows "locally_path_connected_space Y" |
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unfolding locally_path_connected_space_open_path_components |
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proof clarify |
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fix V C |
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assume V: "openin Y V" and c: "C \<in> path_components_of (subtopology Y V)" |
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then have sub: "C \<subseteq> topspace Y" |
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using path_connectedin_path_components_of path_connectedin_subset_topspace path_connectedin_subtopology by blast |
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have "openin X {x \<in> topspace X. f x \<in> C}" |
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proof (subst openin_subopen, clarify) |
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fix x |
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assume x: "x \<in> topspace X" and "f x \<in> C" |
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let ?T = "Collect (path_component_of (subtopology X {z \<in> topspace X. f z \<in> V}) x)" |
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show "\<exists>T. openin X T \<and> x \<in> T \<and> T \<subseteq> {x \<in> topspace X. f x \<in> C}" |
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proof (intro exI conjI) |
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have "\<exists>U. openin X U \<and> ?T \<in> path_components_of (subtopology X U)" |
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proof (intro exI conjI) |
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show "openin X ({z \<in> topspace X. f z \<in> V})" |
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using V f openin_subset quotient_map_def by fastforce |
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show "Collect (path_component_of (subtopology X {z \<in> topspace X. f z \<in> V}) x) |
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\<in> path_components_of (subtopology X {z \<in> topspace X. f z \<in> V})" |
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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) |
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qed |
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with X show "openin X ?T" |
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using locally_path_connected_space_open_path_components by blast |
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show x: "x \<in> ?T" |
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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 |
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by fastforce |
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have "f ` ?T \<subseteq> C" |
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proof (rule path_components_of_maximal) |
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show "C \<in> path_components_of (subtopology Y V)" |
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by (simp add: c) |
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show "path_connectedin (subtopology Y V) (f ` ?T)" |
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proof - |
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have "quotient_map (subtopology X {a \<in> topspace X. f a \<in> V}) (subtopology Y V) f" |
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by (simp add: V f quotient_map_restriction) |
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then show ?thesis |
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by (meson path_connectedin_continuous_map_image path_connectedin_path_component_of quotient_imp_continuous_map) |
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qed |
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show "\<not> disjnt C (f ` ?T)" |
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by (metis (no_types, lifting) \<open>f x \<in> C\<close> x disjnt_iff image_eqI) |
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qed |
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then show "?T \<subseteq> {x \<in> topspace X. f x \<in> C}" |
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by (force simp: path_component_of_equiv) |
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qed |
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qed |
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then show "openin Y C" |
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using f sub by (simp add: quotient_map_def) |
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qed |
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lemma homeomorphic_locally_path_connected_space: |
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assumes "X homeomorphic_space Y" |
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shows "locally_path_connected_space X \<longleftrightarrow> locally_path_connected_space Y" |
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proof - |
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obtain f g where "homeomorphic_maps X Y f g" |
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using assms homeomorphic_space_def by fastforce |
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then show ?thesis |
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by (metis (no_types) homeomorphic_map_def homeomorphic_maps_map locally_path_connected_space_quotient_map_image) |
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qed |
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lemma locally_path_connected_space_retraction_map_image: |
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"\<lbrakk>retraction_map X Y r; locally_path_connected_space X\<rbrakk> |
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\<Longrightarrow> locally_path_connected_space Y" |
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using Abstract_Topology.retraction_imp_quotient_map locally_path_connected_space_quotient_map_image by blast |
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lemma locally_path_connected_space_euclideanreal: "locally_path_connected_space euclideanreal" |
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unfolding locally_path_connected_space_def neighbourhood_base_of |
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proof clarsimp |
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fix W and x :: "real" |
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assume "open W" "x \<in> W" |
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then obtain e where "e > 0" and e: "\<And>x'. \<bar>x' - x\<bar> < e \<longrightarrow> x' \<in> W" |
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by (auto simp: open_real) |
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then show "\<exists>U. open U \<and> (\<exists>V. path_connected V \<and> x \<in> U \<and> U \<subseteq> V \<and> V \<subseteq> W)" |
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by (force intro!: convex_imp_path_connected exI [where x = "{x-e<..<x+e}"]) |
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qed |
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lemma locally_path_connected_space_discrete_topology: |
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"locally_path_connected_space (discrete_topology U)" |
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using locally_path_connected_space_open_path_components by fastforce |
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lemma path_component_eq_connected_component_of: |
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assumes "locally_path_connected_space X" |
|
317 |
shows "(path_component_of_set X x = connected_component_of_set X x)" |
|
318 |
proof (cases "x \<in> topspace X") |
|
319 |
case True |
|
320 |
then show ?thesis |
|
321 |
using connectedin_connected_component_of [of X x] |
|
322 |
apply (clarsimp simp add: connectedin_def connected_space_clopen_in topspace_subtopology_subset cong: conj_cong) |
|
323 |
apply (drule_tac x="path_component_of_set X x" in spec) |
|
324 |
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) |
|
325 |
next |
|
326 |
case False |
|
327 |
then show ?thesis |
|
328 |
using connected_component_of_eq_empty path_component_of_eq_empty by fastforce |
|
329 |
qed |
|
330 |
||
331 |
lemma path_components_eq_connected_components_of: |
|
332 |
"locally_path_connected_space X \<Longrightarrow> (path_components_of X = connected_components_of X)" |
|
333 |
by (simp add: path_components_of_def connected_components_of_def image_def path_component_eq_connected_component_of) |
|
334 |
||
335 |
lemma path_connected_eq_connected_space: |
|
336 |
"locally_path_connected_space X |
|
337 |
\<Longrightarrow> path_connected_space X \<longleftrightarrow> connected_space X" |
|
338 |
by (metis connected_components_of_subset_sing path_components_eq_connected_components_of path_components_of_subset_singleton) |
|
339 |
||
340 |
lemma locally_path_connected_space_product_topology: |
|
341 |
"locally_path_connected_space(product_topology X I) \<longleftrightarrow> |
|
342 |
topspace(product_topology X I) = {} \<or> |
|
343 |
finite {i. i \<in> I \<and> ~path_connected_space(X i)} \<and> |
|
344 |
(\<forall>i \<in> I. locally_path_connected_space(X i))" |
|
345 |
(is "?lhs \<longleftrightarrow> ?empty \<or> ?rhs") |
|
346 |
proof (cases ?empty) |
|
347 |
case True |
|
348 |
then show ?thesis |
|
349 |
by (simp add: locally_path_connected_space_def neighbourhood_base_of openin_closedin_eq) |
|
350 |
next |
|
351 |
case False |
|
352 |
then obtain z where z: "z \<in> (\<Pi>\<^sub>E i\<in>I. topspace (X i))" |
|
353 |
by auto |
|
354 |
have ?rhs if L: ?lhs |
|
355 |
proof - |
|
356 |
obtain U C where U: "openin (product_topology X I) U" |
|
357 |
and V: "path_connectedin (product_topology X I) C" |
|
358 |
and "z \<in> U" "U \<subseteq> C" and Csub: "C \<subseteq> (\<Pi>\<^sub>E i\<in>I. topspace (X i))" |
|
359 |
using L apply (clarsimp simp add: locally_path_connected_space_def neighbourhood_base_of) |
|
360 |
by (metis openin_topspace topspace_product_topology z) |
|
361 |
then obtain V where finV: "finite {i \<in> I. V i \<noteq> topspace (X i)}" |
|
362 |
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" |
|
363 |
by (force simp: openin_product_topology_alt) |
|
364 |
show ?thesis |
|
365 |
proof (intro conjI ballI) |
|
366 |
have "path_connected_space (X i)" if "i \<in> I" "V i = topspace (X i)" for i |
|
367 |
proof - |
|
368 |
have pc: "path_connectedin (X i) ((\<lambda>x. x i) ` C)" |
|
369 |
apply (rule path_connectedin_continuous_map_image [OF _ V]) |
|
370 |
by (simp add: continuous_map_product_projection \<open>i \<in> I\<close>) |
|
371 |
moreover have "((\<lambda>x. x i) ` C) = topspace (X i)" |
|
372 |
proof |
|
373 |
show "(\<lambda>x. x i) ` C \<subseteq> topspace (X i)" |
|
374 |
by (simp add: pc path_connectedin_subset_topspace) |
|
375 |
have "V i \<subseteq> (\<lambda>x. x i) ` (\<Pi>\<^sub>E i\<in>I. V i)" |
|
376 |
by (metis \<open>z \<in> Pi\<^sub>E I V\<close> empty_iff image_projection_PiE order_refl that(1)) |
|
377 |
also have "\<dots> \<subseteq> (\<lambda>x. x i) ` U" |
|
378 |
using subU by blast |
|
379 |
finally show "topspace (X i) \<subseteq> (\<lambda>x. x i) ` C" |
|
380 |
using \<open>U \<subseteq> C\<close> that by blast |
|
381 |
qed |
|
382 |
ultimately show ?thesis |
|
383 |
by (simp add: path_connectedin_topspace) |
|
384 |
qed |
|
385 |
then have "{i \<in> I. \<not> path_connected_space (X i)} \<subseteq> {i \<in> I. V i \<noteq> topspace (X i)}" |
|
386 |
by blast |
|
387 |
with finV show "finite {i \<in> I. \<not> path_connected_space (X i)}" |
|
388 |
using finite_subset by blast |
|
389 |
next |
|
390 |
show "locally_path_connected_space (X i)" if "i \<in> I" for i |
|
391 |
apply (rule locally_path_connected_space_quotient_map_image [OF _ L, where f = "\<lambda>x. x i"]) |
|
392 |
by (metis False Abstract_Topology.retraction_imp_quotient_map retraction_map_product_projection that) |
|
393 |
qed |
|
394 |
qed |
|
395 |
moreover have ?lhs if R: ?rhs |
|
396 |
proof (clarsimp simp add: locally_path_connected_space_def neighbourhood_base_of) |
|
397 |
fix F z |
|
398 |
assume "openin (product_topology X I) F" and "z \<in> F" |
|
399 |
then obtain W where finW: "finite {i \<in> I. W i \<noteq> topspace (X i)}" |
|
400 |
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" |
|
401 |
by (auto simp: openin_product_topology_alt) |
|
402 |
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> |
|
403 |
(W i = topspace (X i) \<and> |
|
404 |
path_connected_space (X i) \<longrightarrow> U = topspace (X i) \<and> C = topspace (X i))" |
|
405 |
(is "\<forall>i \<in> I. ?\<Phi> i") |
|
406 |
proof |
|
407 |
fix i assume "i \<in> I" |
|
408 |
have "locally_path_connected_space (X i)" |
|
409 |
by (simp add: R \<open>i \<in> I\<close>) |
|
410 |
moreover have "openin (X i) (W i) " "z i \<in> W i" |
|
411 |
using \<open>z \<in> Pi\<^sub>E I W\<close> opeW \<open>i \<in> I\<close> by auto |
|
412 |
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" |
|
413 |
using \<open>i \<in> I\<close> by (force simp: locally_path_connected_space_def neighbourhood_base_of) |
|
414 |
show "?\<Phi> i" |
|
415 |
proof (cases "W i = topspace (X i) \<and> path_connected_space(X i)") |
|
416 |
case True |
|
417 |
then show ?thesis |
|
418 |
using \<open>z i \<in> W i\<close> path_connectedin_topspace by blast |
|
419 |
next |
|
420 |
case False |
|
421 |
then show ?thesis |
|
422 |
by (meson UC) |
|
423 |
qed |
|
424 |
qed |
|
425 |
then obtain U C where |
|
426 |
*: "\<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> |
|
427 |
(W i = topspace (X i) \<and> path_connected_space (X i) |
|
428 |
\<longrightarrow> (U i) = topspace (X i) \<and> (C i) = topspace (X i))" |
|
429 |
by metis |
|
430 |
let ?A = "{i \<in> I. \<not> path_connected_space (X i)} \<union> {i \<in> I. W i \<noteq> topspace (X i)}" |
|
431 |
have "{i \<in> I. U i \<noteq> topspace (X i)} \<subseteq> ?A" |
|
432 |
by (clarsimp simp add: "*") |
|
433 |
moreover have "finite ?A" |
|
434 |
by (simp add: that finW) |
|
435 |
ultimately have "finite {i \<in> I. U i \<noteq> topspace (X i)}" |
|
436 |
using finite_subset by auto |
|
437 |
then have "openin (product_topology X I) (Pi\<^sub>E I U)" |
|
438 |
using * by (simp add: openin_PiE_gen) |
|
439 |
then show "\<exists>U. openin (product_topology X I) U \<and> |
|
440 |
(\<exists>V. path_connectedin (product_topology X I) V \<and> z \<in> U \<and> U \<subseteq> V \<and> V \<subseteq> F)" |
|
441 |
apply (rule_tac x="PiE I U" in exI, simp) |
|
442 |
apply (rule_tac x="PiE I C" in exI) |
|
443 |
using \<open>z \<in> Pi\<^sub>E I W\<close> \<open>Pi\<^sub>E I W \<subseteq> F\<close> * |
|
444 |
apply (simp add: path_connectedin_PiE subset_PiE PiE_iff PiE_mono dual_order.trans) |
|
445 |
done |
|
446 |
qed |
|
447 |
ultimately show ?thesis |
|
448 |
using False by blast |
|
449 |
qed |
|
450 |
||
78037
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
451 |
lemma locally_path_connected_is_realinterval: |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
452 |
assumes "is_interval S" |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
453 |
shows "locally_path_connected_space(subtopology euclideanreal S)" |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
454 |
unfolding locally_path_connected_space_def |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
455 |
proof (clarsimp simp add: neighbourhood_base_of openin_subtopology_alt) |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
456 |
fix a U |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
457 |
assume "a \<in> S" and "a \<in> U" and "open U" |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
458 |
then obtain r where "r > 0" and r: "ball a r \<subseteq> U" |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
459 |
by (metis open_contains_ball_eq) |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
460 |
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)" |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
461 |
proof (intro exI conjI) |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
462 |
show "path_connectedin (top_of_set S) (S \<inter> ball a r)" |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
463 |
by (simp add: assms is_interval_Int is_interval_ball_real is_interval_path_connected path_connectedin_subtopology) |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
464 |
show "a \<in> ball a r" |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
465 |
by (simp add: \<open>0 < r\<close>) |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
466 |
qed (use \<open>0 < r\<close> r in auto) |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
467 |
qed |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
468 |
|
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
469 |
lemma locally_path_connected_real_interval: |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
470 |
"locally_path_connected_space (subtopology euclideanreal{a..b})" |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
471 |
"locally_path_connected_space (subtopology euclideanreal{a<..<b})" |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
472 |
using locally_path_connected_is_realinterval |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
473 |
by (auto simp add: is_interval_1) |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
474 |
|
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
475 |
lemma locally_path_connected_space_prod_topology: |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
476 |
"locally_path_connected_space (prod_topology X Y) \<longleftrightarrow> |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
477 |
topspace (prod_topology X Y) = {} \<or> |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
478 |
locally_path_connected_space X \<and> locally_path_connected_space Y" (is "?lhs=?rhs") |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
479 |
proof (cases "topspace(prod_topology X Y) = {}") |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
480 |
case True |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
481 |
then show ?thesis |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
482 |
by (metis equals0D locally_path_connected_space_def neighbourhood_base_of_def) |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
483 |
next |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
484 |
case False |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
485 |
then have ne: "topspace X \<noteq> {}" "topspace Y \<noteq> {}" |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
486 |
by simp_all |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
487 |
show ?thesis |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
488 |
proof |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
489 |
assume ?lhs then show ?rhs |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
490 |
by (metis ne locally_path_connected_space_retraction_map_image retraction_map_fst retraction_map_snd) |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
491 |
next |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
492 |
assume ?rhs |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
493 |
with False have X: "locally_path_connected_space X" and Y: "locally_path_connected_space Y" |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
494 |
by auto |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
495 |
show ?lhs |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
496 |
unfolding locally_path_connected_space_def neighbourhood_base_of |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
497 |
proof clarify |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
498 |
fix UV x y |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
499 |
assume UV: "openin (prod_topology X Y) UV" and "(x,y) \<in> UV" |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
500 |
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" |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
501 |
using X Y by (metis UV \<open>(x,y) \<in> UV\<close> openin_prod_topology_alt) |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
502 |
then obtain C D K L |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
503 |
where "openin X C" "path_connectedin X K" "x \<in> C" "C \<subseteq> K" "K \<subseteq> A" |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
504 |
"openin Y D" "path_connectedin Y L" "y \<in> D" "D \<subseteq> L" "L \<subseteq> B" |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
505 |
by (metis X Y locally_path_connected_space) |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
506 |
with W12 \<open>openin X C\<close> \<open>openin Y D\<close> |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
507 |
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" |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
508 |
apply (rule_tac x="C \<times> D" in exI) |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
509 |
apply (rule_tac x="K \<times> L" in exI) |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
510 |
apply (auto simp: openin_prod_Times_iff path_connectedin_Times) |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
511 |
done |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
512 |
qed |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
513 |
qed |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
514 |
qed |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
515 |
|
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
516 |
lemma locally_path_connected_space_sum_topology: |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
517 |
"locally_path_connected_space(sum_topology X I) \<longleftrightarrow> |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
518 |
(\<forall>i \<in> I. locally_path_connected_space (X i))" (is "?lhs=?rhs") |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
519 |
proof |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
520 |
assume ?lhs then show ?rhs |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
521 |
by (smt (verit) homeomorphic_locally_path_connected_space locally_path_connected_space_open_subset topological_property_of_sum_component) |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
522 |
next |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
523 |
assume R: ?rhs |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
524 |
show ?lhs |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
525 |
proof (clarsimp simp add: locally_path_connected_space_def neighbourhood_base_of forall_openin_sum_topology imp_conjL) |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
526 |
fix W i x |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
527 |
assume ope: "\<forall>i\<in>I. openin (X i) (W i)" |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
528 |
and "i \<in> I" and "x \<in> W i" |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
529 |
then obtain U V where U: "openin (X i) U" and V: "path_connectedin (X i) V" |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
530 |
and "x \<in> U" "U \<subseteq> V" "V \<subseteq> W i" |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
531 |
by (metis R \<open>i \<in> I\<close> \<open>x \<in> W i\<close> locally_path_connected_space) |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
532 |
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)" |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
533 |
proof (intro exI conjI) |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
534 |
show "openin (sum_topology X I) (Pair i ` U)" |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
535 |
by (meson U \<open>i \<in> I\<close> open_map_component_injection open_map_def) |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
536 |
show "path_connectedin (sum_topology X I) (Pair i ` V)" |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
537 |
by (meson V \<open>i \<in> I\<close> continuous_map_component_injection path_connectedin_continuous_map_image) |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
538 |
show "Pair i ` V \<subseteq> Sigma I W" |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
539 |
using \<open>V \<subseteq> W i\<close> \<open>i \<in> I\<close> by force |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
540 |
qed (use \<open>x \<in> U\<close> \<open>U \<subseteq> V\<close> in auto) |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
541 |
qed |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
542 |
qed |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
543 |
|
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
544 |
|
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
545 |
subsection\<open>Locally connected spaces\<close> |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
546 |
|
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
547 |
definition weakly_locally_connected_at |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
548 |
where "weakly_locally_connected_at x X \<equiv> neighbourhood_base_at x (connectedin X) X" |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
549 |
|
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
550 |
definition locally_connected_at |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
551 |
where "locally_connected_at x X \<equiv> |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
552 |
neighbourhood_base_at x (\<lambda>U. openin X U \<and> connectedin X U ) X" |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
553 |
|
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
554 |
definition locally_connected_space |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
555 |
where "locally_connected_space X \<equiv> neighbourhood_base_of (connectedin X) X" |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
556 |
|
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
557 |
|
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
558 |
lemma locally_connected_A: "(\<forall>U x. openin X U \<and> x \<in> U |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
559 |
\<longrightarrow> openin X (connected_component_of_set (subtopology X U) x)) |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
560 |
\<Longrightarrow> neighbourhood_base_of (\<lambda>U. openin X U \<and> connectedin X U) X" |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
561 |
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) |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
562 |
|
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
563 |
lemma locally_connected_B: "locally_connected_space X \<Longrightarrow> |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
564 |
(\<forall>U x. openin X U \<and> x \<in> U \<longrightarrow> openin X (connected_component_of_set (subtopology X U) x))" |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
565 |
unfolding locally_connected_space_def neighbourhood_base_of |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
566 |
apply (erule all_forward) |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
567 |
apply clarify |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
568 |
apply (subst openin_subopen) |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
569 |
by (smt (verit, ccfv_threshold) Ball_Collect connected_component_of_def connected_component_of_equiv connectedin_subtopology in_mono mem_Collect_eq) |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
570 |
|
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
571 |
lemma locally_connected_C: "neighbourhood_base_of (\<lambda>U. openin X U \<and> connectedin X U) X \<Longrightarrow> locally_connected_space X" |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
572 |
using locally_connected_space_def neighbourhood_base_of_mono by auto |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
573 |
|
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
574 |
|
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
575 |
lemma locally_connected_space_alt: |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
576 |
"locally_connected_space X \<longleftrightarrow> neighbourhood_base_of (\<lambda>U. openin X U \<and> connectedin X U) X" |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
577 |
using locally_connected_A locally_connected_B locally_connected_C by blast |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
578 |
|
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
579 |
lemma locally_connected_space_eq_open_connected_component_of: |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
580 |
"locally_connected_space X \<longleftrightarrow> |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
581 |
(\<forall>U x. openin X U \<and> x \<in> U |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
582 |
\<longrightarrow> openin X (connected_component_of_set (subtopology X U) x))" |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
583 |
by (meson locally_connected_A locally_connected_B locally_connected_C) |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
584 |
|
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
585 |
lemma locally_connected_space: |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
586 |
"locally_connected_space X \<longleftrightarrow> |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
587 |
(\<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))" |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
588 |
by (simp add: locally_connected_space_alt open_neighbourhood_base_of) |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
589 |
|
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
590 |
lemma locally_path_connected_imp_locally_connected_space: |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
591 |
"locally_path_connected_space X \<Longrightarrow> locally_connected_space X" |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
592 |
by (simp add: locally_connected_space_def locally_path_connected_space_def neighbourhood_base_of_mono path_connectedin_imp_connectedin) |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
593 |
|
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
594 |
lemma locally_connected_space_open_connected_components: |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
595 |
"locally_connected_space X \<longleftrightarrow> |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
596 |
(\<forall>U C. openin X U \<and> C \<in> connected_components_of(subtopology X U) \<longrightarrow> openin X C)" |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
597 |
apply (simp add: locally_connected_space_eq_open_connected_component_of connected_components_of_def) |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
598 |
by (smt (verit) imageE image_eqI inf.orderE inf_commute openin_subset) |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
599 |
|
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
600 |
lemma openin_connected_component_of_locally_connected_space: |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
601 |
"locally_connected_space X \<Longrightarrow> openin X (connected_component_of_set X x)" |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
602 |
by (metis connected_component_of_eq_empty locally_connected_space_eq_open_connected_component_of openin_empty openin_topspace subtopology_topspace) |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
603 |
|
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
604 |
lemma openin_connected_components_of_locally_connected_space: |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
605 |
"\<lbrakk>locally_connected_space X; C \<in> connected_components_of X\<rbrakk> \<Longrightarrow> openin X C" |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
606 |
by (metis locally_connected_space_open_connected_components openin_topspace subtopology_topspace) |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
607 |
|
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
608 |
lemma weakly_locally_connected_at: |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
609 |
"weakly_locally_connected_at x X \<longleftrightarrow> |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
610 |
(\<forall>V. openin X V \<and> x \<in> V |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
611 |
\<longrightarrow> (\<exists>U. openin X U \<and> x \<in> U \<and> U \<subseteq> V \<and> |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
612 |
(\<forall>y \<in> U. \<exists>C. connectedin X C \<and> C \<subseteq> V \<and> x \<in> C \<and> y \<in> C)))" (is "?lhs=?rhs") |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
613 |
proof |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
614 |
assume ?lhs then show ?rhs |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
615 |
unfolding neighbourhood_base_at_def weakly_locally_connected_at_def |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
616 |
by (meson subsetD subset_trans) |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
617 |
next |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
618 |
assume R: ?rhs |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
619 |
show ?lhs |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
620 |
unfolding neighbourhood_base_at_def weakly_locally_connected_at_def |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
621 |
proof clarify |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
622 |
fix V |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
623 |
assume "openin X V" and "x \<in> V" |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
624 |
then obtain U where "openin X U" "x \<in> U" "U \<subseteq> V" |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
625 |
and U: "\<forall>y\<in>U. \<exists>C. connectedin X C \<and> C \<subseteq> V \<and> x \<in> C \<and> y \<in> C" |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
626 |
using R by force |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
627 |
show "\<exists>A B. openin X A \<and> connectedin X B \<and> x \<in> A \<and> A \<subseteq> B \<and> B \<subseteq> V" |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
628 |
proof (intro conjI exI) |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
629 |
show "connectedin X (connected_component_of_set (subtopology X V) x)" |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
630 |
by (meson connectedin_connected_component_of connectedin_subtopology) |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
631 |
show "U \<subseteq> connected_component_of_set (subtopology X V) x" |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
632 |
using connected_component_of_maximal U |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
633 |
by (simp add: connected_component_of_def connectedin_subtopology subsetI) |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
634 |
show "connected_component_of_set (subtopology X V) x \<subseteq> V" |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
635 |
using connected_component_of_subset_topspace by fastforce |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
636 |
qed (auto simp: \<open>x \<in> U\<close> \<open>openin X U\<close>) |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
637 |
qed |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
638 |
qed |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
639 |
|
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
640 |
lemma locally_connected_space_iff_weak: |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
641 |
"locally_connected_space X \<longleftrightarrow> (\<forall>x \<in> topspace X. weakly_locally_connected_at x X)" |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
642 |
by (simp add: locally_connected_space_def neighbourhood_base_of_def weakly_locally_connected_at_def) |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
643 |
|
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
644 |
lemma locally_connected_space_im_kleinen: |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
645 |
"locally_connected_space X \<longleftrightarrow> |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
646 |
(\<forall>V x. openin X V \<and> x \<in> V |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
647 |
\<longrightarrow> (\<exists>U. openin X U \<and> x \<in> U \<and> U \<subseteq> V \<and> |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
648 |
(\<forall>y \<in> U. \<exists>C. connectedin X C \<and> C \<subseteq> V \<and> x \<in> C \<and> y \<in> C)))" |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
649 |
unfolding locally_connected_space_iff_weak weakly_locally_connected_at |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
650 |
using openin_subset subsetD by fastforce |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
651 |
|
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
652 |
lemma locally_connected_space_open_subset: |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
653 |
"\<lbrakk>locally_connected_space X; openin X S\<rbrakk> \<Longrightarrow> locally_connected_space (subtopology X S)" |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
654 |
apply (simp add: locally_connected_space_def) |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
655 |
by (smt (verit, ccfv_threshold) connectedin_subtopology neighbourhood_base_of openin_open_subtopology subset_trans) |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
656 |
|
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
657 |
lemma locally_connected_space_quotient_map_image: |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
658 |
assumes X: "locally_connected_space X" and f: "quotient_map X Y f" |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
659 |
shows "locally_connected_space Y" |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
660 |
unfolding locally_connected_space_open_connected_components |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
661 |
proof clarify |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
662 |
fix V C |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
663 |
assume "openin Y V" and C: "C \<in> connected_components_of (subtopology Y V)" |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
664 |
then have "C \<subseteq> topspace Y" |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
665 |
using connected_components_of_subset by force |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
666 |
have ope1: "openin X {a \<in> topspace X. f a \<in> V}" |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
667 |
using \<open>openin Y V\<close> f openin_continuous_map_preimage quotient_imp_continuous_map by blast |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
668 |
define Vf where "Vf \<equiv> {z \<in> topspace X. f z \<in> V}" |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
669 |
have "openin X {x \<in> topspace X. f x \<in> C}" |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
670 |
proof (clarsimp simp: openin_subopen [where S = "{x \<in> topspace X. f x \<in> C}"]) |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
671 |
fix x |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
672 |
assume "x \<in> topspace X" and "f x \<in> C" |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
673 |
show "\<exists>T. openin X T \<and> x \<in> T \<and> T \<subseteq> {x \<in> topspace X. f x \<in> C}" |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
674 |
proof (intro exI conjI) |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
675 |
show "openin X (connected_component_of_set (subtopology X Vf) x)" |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
676 |
by (metis Vf_def X connected_component_of_eq_empty locally_connected_B ope1 openin_empty |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
677 |
openin_subset topspace_subtopology_subset) |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
678 |
show x_in_conn: "x \<in> connected_component_of_set (subtopology X Vf) x" |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
679 |
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 |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
680 |
have "connected_component_of_set (subtopology X Vf) x \<subseteq> topspace X \<inter> Vf" |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
681 |
using connected_component_of_subset_topspace by fastforce |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
682 |
moreover |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
683 |
have "f ` connected_component_of_set (subtopology X Vf) x \<subseteq> C" |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
684 |
proof (rule connected_components_of_maximal [where X = "subtopology Y V"]) |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
685 |
show "C \<in> connected_components_of (subtopology Y V)" |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
686 |
by (simp add: C) |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
687 |
have \<section>: "quotient_map (subtopology X Vf) (subtopology Y V) f" |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
688 |
by (simp add: Vf_def \<open>openin Y V\<close> f quotient_map_restriction) |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
689 |
then show "connectedin (subtopology Y V) (f ` connected_component_of_set (subtopology X Vf) x)" |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
690 |
by (metis connectedin_connected_component_of connectedin_continuous_map_image quotient_imp_continuous_map) |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
691 |
show "\<not> disjnt C (f ` connected_component_of_set (subtopology X Vf) x)" |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
692 |
using \<open>f x \<in> C\<close> x_in_conn by (auto simp: disjnt_iff) |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
693 |
qed |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
694 |
ultimately |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
695 |
show "connected_component_of_set (subtopology X Vf) x \<subseteq> {x \<in> topspace X. f x \<in> C}" |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
696 |
by blast |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
697 |
qed |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
698 |
qed |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
699 |
then show "openin Y C" |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
700 |
using \<open>C \<subseteq> topspace Y\<close> f quotient_map_def by fastforce |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
701 |
qed |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
702 |
|
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
703 |
|
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
704 |
lemma locally_connected_space_retraction_map_image: |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
705 |
"\<lbrakk>retraction_map X Y r; locally_connected_space X\<rbrakk> |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
706 |
\<Longrightarrow> locally_connected_space Y" |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
707 |
using locally_connected_space_quotient_map_image retraction_imp_quotient_map by blast |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
708 |
|
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
709 |
lemma homeomorphic_locally_connected_space: |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
710 |
"X homeomorphic_space Y \<Longrightarrow> locally_connected_space X \<longleftrightarrow> locally_connected_space Y" |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
711 |
by (meson homeomorphic_map_def homeomorphic_space homeomorphic_space_sym locally_connected_space_quotient_map_image) |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
712 |
|
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
713 |
lemma locally_connected_space_euclideanreal: "locally_connected_space euclideanreal" |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
714 |
by (simp add: locally_path_connected_imp_locally_connected_space locally_path_connected_space_euclideanreal) |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
715 |
|
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
716 |
lemma locally_connected_is_realinterval: |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
717 |
"is_interval S \<Longrightarrow> locally_connected_space(subtopology euclideanreal S)" |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
718 |
by (simp add: locally_path_connected_imp_locally_connected_space locally_path_connected_is_realinterval) |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
719 |
|
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
720 |
lemma locally_connected_real_interval: |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
721 |
"locally_connected_space (subtopology euclideanreal{a..b})" |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
722 |
"locally_connected_space (subtopology euclideanreal{a<..<b})" |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
723 |
using connected_Icc is_interval_connected_1 locally_connected_is_realinterval by auto |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
724 |
|
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
725 |
lemma locally_connected_space_discrete_topology: |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
726 |
"locally_connected_space (discrete_topology U)" |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
727 |
by (simp add: locally_path_connected_imp_locally_connected_space locally_path_connected_space_discrete_topology) |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
728 |
|
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
729 |
lemma locally_path_connected_imp_locally_connected_at: |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
730 |
"locally_path_connected_at x X \<Longrightarrow> locally_connected_at x X" |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
731 |
by (simp add: locally_connected_at_def locally_path_connected_at_def neighbourhood_base_at_mono path_connectedin_imp_connectedin) |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
732 |
|
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
733 |
lemma weakly_locally_path_connected_imp_weakly_locally_connected_at: |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
734 |
"weakly_locally_path_connected_at x X |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
735 |
\<Longrightarrow> weakly_locally_connected_at x X" |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
736 |
by (simp add: neighbourhood_base_at_mono path_connectedin_imp_connectedin weakly_locally_connected_at_def weakly_locally_path_connected_at_def) |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
737 |
|
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
738 |
|
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
739 |
lemma interior_of_locally_connected_subspace_component: |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
740 |
assumes X: "locally_connected_space X" |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
741 |
and C: "C \<in> connected_components_of (subtopology X S)" |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
742 |
shows "X interior_of C = C \<inter> X interior_of S" |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
743 |
proof - |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
744 |
obtain Csub: "C \<subseteq> topspace X" "C \<subseteq> S" |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
745 |
by (meson C connectedin_connected_components_of connectedin_subset_topspace connectedin_subtopology) |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
746 |
show ?thesis |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
747 |
proof |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
748 |
show "X interior_of C \<subseteq> C \<inter> X interior_of S" |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
749 |
by (simp add: Csub interior_of_mono interior_of_subset) |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
750 |
have eq: "X interior_of S = \<Union> (connected_components_of (subtopology X (X interior_of S)))" |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
751 |
by (metis Union_connected_components_of interior_of_subset_topspace topspace_subtopology_subset) |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
752 |
moreover have "C \<inter> D \<subseteq> X interior_of C" |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
753 |
if "D \<in> connected_components_of (subtopology X (X interior_of S))" for D |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
754 |
proof (cases "C \<inter> D = {}") |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
755 |
case False |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
756 |
have "D \<subseteq> X interior_of C" |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
757 |
proof (rule interior_of_maximal) |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
758 |
have "connectedin (subtopology X S) D" |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
759 |
by (meson connectedin_connected_components_of connectedin_subtopology interior_of_subset subset_trans that) |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
760 |
then show "D \<subseteq> C" |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
761 |
by (meson C False connected_components_of_maximal disjnt_def) |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
762 |
show "openin X D" |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
763 |
using X locally_connected_space_open_connected_components openin_interior_of that by blast |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
764 |
qed |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
765 |
then show ?thesis |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
766 |
by blast |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
767 |
qed auto |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
768 |
ultimately show "C \<inter> X interior_of S \<subseteq> X interior_of C" |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
769 |
by blast |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
770 |
qed |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
771 |
qed |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
772 |
|
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
773 |
|
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
774 |
lemma frontier_of_locally_connected_subspace_component: |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
775 |
assumes X: "locally_connected_space X" and "closedin X S" |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
776 |
and C: "C \<in> connected_components_of (subtopology X S)" |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
777 |
shows "X frontier_of C = C \<inter> X frontier_of S" |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
778 |
proof - |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
779 |
obtain Csub: "C \<subseteq> topspace X" "C \<subseteq> S" |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
780 |
by (meson C connectedin_connected_components_of connectedin_subset_topspace connectedin_subtopology) |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
781 |
then have "X closure_of C - X interior_of C = C \<inter> X closure_of S - C \<inter> X interior_of S" |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
782 |
using assms |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
783 |
apply (simp add: closure_of_closedin flip: interior_of_locally_connected_subspace_component) |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
784 |
by (metis closedin_connected_components_of closedin_trans_full closure_of_eq inf.orderE) |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
785 |
then show ?thesis |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
786 |
by (simp add: Diff_Int_distrib frontier_of_def) |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
787 |
qed |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
788 |
|
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
789 |
(*Similar proof to locally_connected_space_prod_topology*) |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
790 |
lemma locally_connected_space_prod_topology: |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
791 |
"locally_connected_space (prod_topology X Y) \<longleftrightarrow> |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
792 |
topspace (prod_topology X Y) = {} \<or> |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
793 |
locally_connected_space X \<and> locally_connected_space Y" (is "?lhs=?rhs") |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
794 |
proof (cases "topspace(prod_topology X Y) = {}") |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
795 |
case True |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
796 |
then show ?thesis |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
797 |
using locally_connected_space_iff_weak by force |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
798 |
next |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
799 |
case False |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
800 |
then have ne: "topspace X \<noteq> {}" "topspace Y \<noteq> {}" |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
801 |
by simp_all |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
802 |
show ?thesis |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
803 |
proof |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
804 |
assume ?lhs then show ?rhs |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
805 |
by (metis locally_connected_space_quotient_map_image ne quotient_map_fst quotient_map_snd) |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
806 |
next |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
807 |
assume ?rhs |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
808 |
with False have X: "locally_connected_space X" and Y: "locally_connected_space Y" |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
809 |
by auto |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
810 |
show ?lhs |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
811 |
unfolding locally_connected_space_def neighbourhood_base_of |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
812 |
proof clarify |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
813 |
fix UV x y |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
814 |
assume UV: "openin (prod_topology X Y) UV" and "(x,y) \<in> UV" |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
815 |
|
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
816 |
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" |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
817 |
using X Y by (metis UV \<open>(x,y) \<in> UV\<close> openin_prod_topology_alt) |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
818 |
then obtain C D K L |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
819 |
where "openin X C" "connectedin X K" "x \<in> C" "C \<subseteq> K" "K \<subseteq> A" |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
820 |
"openin Y D" "connectedin Y L" "y \<in> D" "D \<subseteq> L" "L \<subseteq> B" |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
821 |
by (metis X Y locally_connected_space) |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
822 |
with W12 \<open>openin X C\<close> \<open>openin Y D\<close> |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
823 |
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" |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
824 |
apply (rule_tac x="C \<times> D" in exI) |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
825 |
apply (rule_tac x="K \<times> L" in exI) |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
826 |
apply (auto simp: openin_prod_Times_iff connectedin_Times) |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
827 |
done |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
828 |
qed |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
829 |
qed |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
830 |
qed |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
831 |
|
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
832 |
(*Same proof as locally_path_connected_space_product_topology*) |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
833 |
lemma locally_connected_space_product_topology: |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
834 |
"locally_connected_space(product_topology X I) \<longleftrightarrow> |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
835 |
topspace(product_topology X I) = {} \<or> |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
836 |
finite {i. i \<in> I \<and> ~connected_space(X i)} \<and> |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
837 |
(\<forall>i \<in> I. locally_connected_space(X i))" |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
838 |
(is "?lhs \<longleftrightarrow> ?empty \<or> ?rhs") |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
839 |
proof (cases ?empty) |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
840 |
case True |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
841 |
then show ?thesis |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
842 |
by (simp add: locally_connected_space_def neighbourhood_base_of openin_closedin_eq) |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
843 |
next |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
844 |
case False |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
845 |
then obtain z where z: "z \<in> (\<Pi>\<^sub>E i\<in>I. topspace (X i))" |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
846 |
by auto |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
847 |
have ?rhs if L: ?lhs |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
848 |
proof - |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
849 |
obtain U C where U: "openin (product_topology X I) U" |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
850 |
and V: "connectedin (product_topology X I) C" |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
851 |
and "z \<in> U" "U \<subseteq> C" and Csub: "C \<subseteq> (\<Pi>\<^sub>E i\<in>I. topspace (X i))" |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
852 |
using L apply (clarsimp simp add: locally_connected_space_def neighbourhood_base_of) |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
853 |
by (metis openin_topspace topspace_product_topology z) |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
854 |
then obtain V where finV: "finite {i \<in> I. V i \<noteq> topspace (X i)}" |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
855 |
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" |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
856 |
by (force simp: openin_product_topology_alt) |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
857 |
show ?thesis |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
858 |
proof (intro conjI ballI) |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
859 |
have "connected_space (X i)" if "i \<in> I" "V i = topspace (X i)" for i |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
860 |
proof - |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
861 |
have pc: "connectedin (X i) ((\<lambda>x. x i) ` C)" |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
862 |
apply (rule connectedin_continuous_map_image [OF _ V]) |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
863 |
by (simp add: continuous_map_product_projection \<open>i \<in> I\<close>) |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
864 |
moreover have "((\<lambda>x. x i) ` C) = topspace (X i)" |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
865 |
proof |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
866 |
show "(\<lambda>x. x i) ` C \<subseteq> topspace (X i)" |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
867 |
by (simp add: pc connectedin_subset_topspace) |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
868 |
have "V i \<subseteq> (\<lambda>x. x i) ` (\<Pi>\<^sub>E i\<in>I. V i)" |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
869 |
by (metis \<open>z \<in> Pi\<^sub>E I V\<close> empty_iff image_projection_PiE order_refl that(1)) |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
870 |
also have "\<dots> \<subseteq> (\<lambda>x. x i) ` U" |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
871 |
using subU by blast |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
872 |
finally show "topspace (X i) \<subseteq> (\<lambda>x. x i) ` C" |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
873 |
using \<open>U \<subseteq> C\<close> that by blast |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
874 |
qed |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
875 |
ultimately show ?thesis |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
876 |
by (simp add: connectedin_topspace) |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
877 |
qed |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
878 |
then have "{i \<in> I. \<not> connected_space (X i)} \<subseteq> {i \<in> I. V i \<noteq> topspace (X i)}" |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
879 |
by blast |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
880 |
with finV show "finite {i \<in> I. \<not> connected_space (X i)}" |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
881 |
using finite_subset by blast |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
882 |
next |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
883 |
show "locally_connected_space (X i)" if "i \<in> I" for i |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
884 |
by (meson False L locally_connected_space_quotient_map_image quotient_map_product_projection that) |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
885 |
qed |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
886 |
qed |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
887 |
moreover have ?lhs if R: ?rhs |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
888 |
proof (clarsimp simp add: locally_connected_space_def neighbourhood_base_of) |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
889 |
fix F z |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
890 |
assume "openin (product_topology X I) F" and "z \<in> F" |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
891 |
then obtain W where finW: "finite {i \<in> I. W i \<noteq> topspace (X i)}" |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
892 |
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" |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
893 |
by (auto simp: openin_product_topology_alt) |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
894 |
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> |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
895 |
(W i = topspace (X i) \<and> |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
896 |
connected_space (X i) \<longrightarrow> U = topspace (X i) \<and> C = topspace (X i))" |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
897 |
(is "\<forall>i \<in> I. ?\<Phi> i") |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
898 |
proof |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
899 |
fix i assume "i \<in> I" |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
900 |
have "locally_connected_space (X i)" |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
901 |
by (simp add: R \<open>i \<in> I\<close>) |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
902 |
moreover have "openin (X i) (W i) " "z i \<in> W i" |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
903 |
using \<open>z \<in> Pi\<^sub>E I W\<close> opeW \<open>i \<in> I\<close> by auto |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
904 |
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" |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
905 |
using \<open>i \<in> I\<close> by (force simp: locally_connected_space_def neighbourhood_base_of) |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
906 |
show "?\<Phi> i" |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
907 |
proof (cases "W i = topspace (X i) \<and> connected_space(X i)") |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
908 |
case True |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
909 |
then show ?thesis |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
910 |
using \<open>z i \<in> W i\<close> connectedin_topspace by blast |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
911 |
next |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
912 |
case False |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
913 |
then show ?thesis |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
914 |
by (meson UC) |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
915 |
qed |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
916 |
qed |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
917 |
then obtain U C where |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
918 |
*: "\<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> |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
919 |
(W i = topspace (X i) \<and> connected_space (X i) |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
920 |
\<longrightarrow> (U i) = topspace (X i) \<and> (C i) = topspace (X i))" |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
921 |
by metis |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
922 |
let ?A = "{i \<in> I. \<not> connected_space (X i)} \<union> {i \<in> I. W i \<noteq> topspace (X i)}" |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
923 |
have "{i \<in> I. U i \<noteq> topspace (X i)} \<subseteq> ?A" |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
924 |
by (clarsimp simp add: "*") |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
925 |
moreover have "finite ?A" |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
926 |
by (simp add: that finW) |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
927 |
ultimately have "finite {i \<in> I. U i \<noteq> topspace (X i)}" |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
928 |
using finite_subset by auto |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
929 |
then have "openin (product_topology X I) (Pi\<^sub>E I U)" |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
930 |
using * by (simp add: openin_PiE_gen) |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
931 |
then show "\<exists>U. openin (product_topology X I) U \<and> |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
932 |
(\<exists>V. connectedin (product_topology X I) V \<and> z \<in> U \<and> U \<subseteq> V \<and> V \<subseteq> F)" |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
933 |
apply (rule_tac x="PiE I U" in exI, simp) |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
934 |
apply (rule_tac x="PiE I C" in exI) |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
935 |
using \<open>z \<in> Pi\<^sub>E I W\<close> \<open>Pi\<^sub>E I W \<subseteq> F\<close> * |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
936 |
apply (simp add: connectedin_PiE subset_PiE PiE_iff PiE_mono dual_order.trans) |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
937 |
done |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
938 |
qed |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
939 |
ultimately show ?thesis |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
940 |
using False by blast |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
941 |
qed |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
942 |
|
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
943 |
lemma locally_connected_space_sum_topology: |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
944 |
"locally_connected_space(sum_topology X I) \<longleftrightarrow> |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
945 |
(\<forall>i \<in> I. locally_connected_space (X i))" (is "?lhs=?rhs") |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
946 |
proof |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
947 |
assume ?lhs then show ?rhs |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
948 |
by (smt (verit) homeomorphic_locally_connected_space locally_connected_space_open_subset topological_property_of_sum_component) |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
949 |
next |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
950 |
assume R: ?rhs |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
951 |
show ?lhs |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
952 |
proof (clarsimp simp add: locally_connected_space_def neighbourhood_base_of forall_openin_sum_topology imp_conjL) |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
953 |
fix W i x |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
954 |
assume ope: "\<forall>i\<in>I. openin (X i) (W i)" |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
955 |
and "i \<in> I" and "x \<in> W i" |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
956 |
then obtain U V where U: "openin (X i) U" and V: "connectedin (X i) V" |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
957 |
and "x \<in> U" "U \<subseteq> V" "V \<subseteq> W i" |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
958 |
by (metis R \<open>i \<in> I\<close> \<open>x \<in> W i\<close> locally_connected_space) |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
959 |
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)" |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
960 |
proof (intro exI conjI) |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
961 |
show "openin (sum_topology X I) (Pair i ` U)" |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
962 |
by (meson U \<open>i \<in> I\<close> open_map_component_injection open_map_def) |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
963 |
show "connectedin (sum_topology X I) (Pair i ` V)" |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
964 |
by (meson V \<open>i \<in> I\<close> continuous_map_component_injection connectedin_continuous_map_image) |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
965 |
show "Pair i ` V \<subseteq> Sigma I W" |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
966 |
using \<open>V \<subseteq> W i\<close> \<open>i \<in> I\<close> by force |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
967 |
qed (use \<open>x \<in> U\<close> \<open>U \<subseteq> V\<close> in auto) |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
968 |
qed |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
969 |
qed |
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
970 |
|
37894dff0111
More material from the HOL Light metric space library
paulson <lp15@cam.ac.uk>
parents:
71172
diff
changeset
|
971 |
|
69945 | 972 |
end |