Quelle  Locally.thy

section ‹Neighbourhood bases and Locally path-connected spaces›

theory Locally
  imports
    Path_Connected Function_Topology Sum_Topology
begin

subsection‹Neighbourhood Bases›

text ‹Useful for "local" properties of various kinds›

definition neighbourhood_base_at where
 "neighbourhood_base_at x P X ≡
        ∀W. openin X W ∧ x ∈ W
            ⟶ (∃U V. openin X U ∧ P V ∧ x ∈ U ∧ U ⊆ V ∧ V ⊆ W)"

definition neighbourhood_base_of where
 "neighbourhood_base_of P X ≡
        ∀x ∈ topspace X. neighbourhood_base_at x P X"

lemma neighbourhood_base_of:
   "neighbourhood_base_of P X ⟷
        (∀W x. openin X W ∧ x ∈ W
          ⟶ (∃U V. openin X U ∧ P V ∧ x ∈ U ∧ U ⊆ V ∧ V ⊆ W))"
  unfolding neighbourhood_base_at_def neighbourhood_base_of_def
  using openin_subset by blast

lemma neighbourhood_base_at_mono:
   "[neighbourhood_base_at x P X; ∧S. [P S; x ∈ S] ==> Q S] ==> neighbourhood_base_at x Q X"
  unfolding neighbourhood_base_at_def
  by (meson subset_eq)

lemma neighbourhood_base_of_mono:
   "[neighbourhood_base_of P X; ∧S. P S ==> Q S] ==> neighbourhood_base_of Q X"
  unfolding neighbourhood_base_of_def
  using neighbourhood_base_at_mono by force

lemma open_neighbourhood_base_at:
   "(∧S. [P S; x ∈ S] ==> openin X S)
        ==> neighbourhood_base_at x P X ⟷ (∀W. openin X W ∧ x ∈ W ⟶ (∃U. P U ∧ x ∈ U ∧ U ⊆ W))"
  unfolding neighbourhood_base_at_def
  by (meson subsetD)

lemma open_neighbourhood_base_of:
  "(∀S. P S ⟶ openin X S)
        ==> neighbourhood_base_of P X ⟷ (∀W x. openin X W ∧ x ∈ W ⟶ (∃U. P U ∧ x ∈ U ∧ U ⊆ W))"
  by (smt (verit) neighbourhood_base_of subsetD)

lemma neighbourhood_base_of_open_subset:
   "[neighbourhood_base_of P X; openin X S]
        ==> neighbourhood_base_of P (subtopology X S)"
  by (smt (verit, ccfv_SIG) neighbourhood_base_of openin_open_subtopology subset_trans)

lemma neighbourhood_base_of_topology_base:
   "openin X = arbitrary union_of (λW. W ∈ B)
        ==> neighbourhood_base_of P X ⟷
             (∀W x. W ∈ B ∧ x ∈ W ⟶ (∃U V. openin X U ∧ P V ∧ x ∈ U ∧ U ⊆ V ∧ V ⊆ W))"
  by (smt (verit, del_insts) neighbourhood_base_of openin_topology_base_unique subset_trans)

lemma neighbourhood_base_at_unlocalized:
  assumes "∧S T. [P S; openin X T; x ∈ T; T ⊆ S] ==> P T"
  shows "neighbourhood_base_at x P X
     ⟷ (x ∈ topspace X ⟶ (∃U V. openin X U ∧ P V ∧ x ∈ U ∧ U ⊆ V ∧ V ⊆ topspace X))"
         (is "?lhs = ?rhs")
proof
  assume R: ?rhs show ?lhs
    unfolding neighbourhood_base_at_def
  proof clarify
    fix W
    assume "openin X W" "x ∈ W"
    then have "x ∈ topspace X"
      using openin_subset by blast
    with R obtain U V where "openin X U" "P V" "x ∈ U" "U ⊆ V" "V ⊆ topspace X"
      by blast
    then show "∃U V. openin X U ∧ P V ∧ x ∈ U ∧ U ⊆ V ∧ V ⊆ W"
      by (metis IntI ‹openin X W› ‹x ∈ W› assms inf_le1 inf_le2 openin_Int)
  qed
qed (simp add: neighbourhood_base_at_def)

lemma neighbourhood_base_at_with_subset:
   "[openin X U; x ∈ U]
        ==> (neighbourhood_base_at x P X ⟷ neighbourhood_base_at x (λT. T ⊆ U ∧ P T) X)"
  unfolding neighbourhood_base_at_def by (metis IntI Int_subset_iff openin_Int)

lemma neighbourhood_base_of_with_subset:
   "neighbourhood_base_of P X ⟷ neighbourhood_base_of (λt. t ⊆ topspace X ∧ P t) X"
  using neighbourhood_base_at_with_subset
  by (fastforce simp add: neighbourhood_base_of_def)

subsection‹Locally path-connected spaces›

definition weakly_locally_path_connected_at
  where "weakly_locally_path_connected_at x X ≡ neighbourhood_base_at x (path_connectedin X) X"

definition locally_path_connected_at
  where "locally_path_connected_at x X ≡
    neighbourhood_base_at x (λU. openin X U ∧ path_connectedin X U) X"

definition locally_path_connected_space
  where "locally_path_connected_space X ≡ neighbourhood_base_of (path_connectedin X) X"

lemma locally_path_connected_space_alt:
  "locally_path_connected_space X ⟷ neighbourhood_base_of (λU. openin X U ∧ path_connectedin X U) X"
  (is "?P = ?Q")
  and locally_path_connected_space_eq_open_path_component_of:
  "locally_path_connected_space X ⟷
        (∀U x. openin X U ∧ x ∈ U ⟶ openin X (Collect (path_component_of (subtopology X U) x)))"
  (is "?P = ?R")
proof -
  have ?P if ?Q
    using locally_path_connected_space_def neighbourhood_base_of_mono that by auto
  moreover have ?R if P: ?P
  proof -
    show ?thesis
    proof clarify
      fix U y
      assume "openin X U" "y ∈ U"
      have "∃T. openin X T ∧ x ∈ T ∧ T ⊆ Collect (path_component_of (subtopology X U) y)"
        if "path_component_of (subtopology X U) y x" for x

      proof -
        have "x ∈ U"
          using path_component_of_equiv that topspace_subtopology by fastforce
        then have "∃Ua. openin X Ua ∧ (∃V. path_connectedin X V ∧ x ∈ Ua ∧ Ua ⊆ V ∧ V ⊆ U)"
      using P
      by (simp add: ‹openin X U› locally_path_connected_space_def neighbourhood_base_of)
        then show ?thesis
          by (metis dual_order.trans path_component_of_equiv path_component_of_maximal path_connectedin_subtopology subset_iff that)
      qed
      then show "openin X (Collect (path_component_of (subtopology X U) y))"
        using openin_subopen by force
    qed
  qed
  moreover have ?Q if ?R
    by (smt (verit) mem_Collect_eq open_neighbourhood_base_of openin_subset path_component_of_refl 
        path_connectedin_path_component_of path_connectedin_subtopology that topspace_subtopology_subset)
  ultimately show "?P = ?Q" "?P = ?R"
    by blast+
qed

lemma locally_path_connected_space:
   "locally_path_connected_space X
   ⟷ (∀V x. openin X V ∧ x ∈ V ⟶ (∃U. openin X U ∧ path_connectedin X U ∧ x ∈ U ∧ U ⊆ V))"
  by (simp add: locally_path_connected_space_alt open_neighbourhood_base_of)

lemma locally_path_connected_space_open_path_components:
   "locally_path_connected_space X ⟷
        (∀U C. openin X U ∧ C ∈ path_components_of(subtopology X U) ⟶ openin X C)"
  apply (simp add: locally_path_connected_space_eq_open_path_component_of path_components_of_def)
  by (smt (verit, ccfv_threshold) Int_iff image_iff openin_subset subset_iff)

lemma openin_path_component_of_locally_path_connected_space:
   "locally_path_connected_space X ==> openin X (Collect (path_component_of X x))"
  using locally_path_connected_space_eq_open_path_component_of openin_subopen path_component_of_eq_empty 
  by fastforce

lemma openin_path_components_of_locally_path_connected_space:
   "[locally_path_connected_space X; C ∈ path_components_of X] ==> openin X C"
  using locally_path_connected_space_open_path_components by force

lemma closedin_path_components_of_locally_path_connected_space:
   "[locally_path_connected_space X; C ∈ path_components_of X] ==> closedin X C"
  unfolding closedin_def
  by (metis Diff_iff complement_path_components_of_Union openin_clauses(3) openin_closedin_eq 
      openin_path_components_of_locally_path_connected_space)

lemma closedin_path_component_of_locally_path_connected_space:
  assumes "locally_path_connected_space X"
  shows "closedin X (Collect (path_component_of X x))"
proof (cases "x ∈ topspace X")
  case True
  then show ?thesis
    by (simp add: assms closedin_path_components_of_locally_path_connected_space path_component_in_path_components_of)
next
  case False
  then show ?thesis
    by (metis closedin_empty path_component_of_eq_empty)
qed

lemma weakly_locally_path_connected_at:
   "weakly_locally_path_connected_at x X ⟷
    (∀V. openin X V ∧ x ∈ V
          ⟶ (∃U. openin X U ∧ x ∈ U ∧ U ⊆ V ∧
                  (∀y ∈ U. ∃C. path_connectedin X C ∧ C ⊆ V ∧ x ∈ C ∧ y ∈ C)))"
         (is "?lhs = ?rhs")
proof
  assume ?lhs then show ?rhs
    by (smt (verit) neighbourhood_base_at_def subset_iff weakly_locally_path_connected_at_def)
next
  have *: "∃V. path_connectedin X V ∧ U ⊆ V ∧ V ⊆ W"
    if "(∀y∈U. ∃C. path_connectedin X C ∧ C ⊆ W ∧ x ∈ C ∧ y ∈ C)"
    for W U
  proof (intro exI conjI)
    let ?V = "(Collect (path_component_of (subtopology X W) x))"
      show "path_connectedin X (Collect (path_component_of (subtopology X W) x))"
        by (meson path_connectedin_path_component_of path_connectedin_subtopology)
      show "U ⊆ ?V"
        by (auto simp: path_component_of path_connectedin_subtopology that)
      show "?V ⊆ W"
        by (meson path_connectedin_path_component_of path_connectedin_subtopology)
    qed
  assume ?rhs
  then show ?lhs
    unfolding weakly_locally_path_connected_at_def neighbourhood_base_at_def by (metis "*")
qed

lemma locally_path_connected_space_im_kleinen:
   "locally_path_connected_space X ⟷
      (∀V x. openin X V ∧ x ∈ V
             ⟶ (∃U. openin X U ∧
                    x ∈ U ∧ U ⊆ V ∧
                    (∀y ∈ U. ∃C. path_connectedin X C ∧
                                C ⊆ V ∧ x ∈ C ∧ y ∈ C)))"
         (is "?lhs = ?rhs")
proof
  show "?lhs ==> ?rhs"
    by (metis locally_path_connected_space)
  assume ?rhs
  then show ?lhs
    unfolding locally_path_connected_space_def neighbourhood_base_of
    by (metis neighbourhood_base_at_def weakly_locally_path_connected_at weakly_locally_path_connected_at_def)
qed

lemma locally_path_connected_space_open_subset:
   "[locally_path_connected_space X; openin X S]
        ==> locally_path_connected_space (subtopology X S)"
  by (smt (verit, best) locally_path_connected_space openin_open_subtopology path_connectedin_subtopology subset_trans)

lemma locally_path_connected_space_quotient_map_image:
  assumes f: "quotient_map X Y f" and X: "locally_path_connected_space X"
  shows "locally_path_connected_space Y"
  unfolding locally_path_connected_space_open_path_components
proof clarify
  fix V C
  assume V: "openin Y V" and c: "C ∈ path_components_of (subtopology Y V)"
  then have sub: "C ⊆ topspace Y"
    using path_connectedin_path_components_of path_connectedin_subset_topspace path_connectedin_subtopology by blast
  have "openin X {x ∈ topspace X. f x ∈ C}"
  proof (subst openin_subopen, clarify)
    fix x
    assume x: "x ∈ topspace X" and "f x ∈ C"
    let ?T = "Collect (path_component_of (subtopology X {z ∈ topspace X. f z ∈ V}) x)"
    show "∃T. openin X T ∧ x ∈ T ∧ T ⊆ {x ∈ topspace X. f x ∈ C}"
    proof (intro exI conjI)
      have *: "∃U. openin X U ∧ ?T ∈ path_components_of (subtopology X U)"
      proof (intro exI conjI)
        show "openin X ({z ∈ topspace X. f z ∈ V})"
          using V f openin_subset quotient_map_def by fastforce
        have "x ∈ topspace (subtopology X {z ∈ topspace X. f z ∈ V})"
          using ‹f x ∈ C› c path_components_of_subset x by force
        then show "Collect (path_component_of (subtopology X {z ∈ topspace X. f z ∈ V}) x)
            ∈ path_components_of (subtopology X {z ∈ topspace X. f z ∈ V})"
          by (meson path_component_in_path_components_of)
      qed
      with X show "openin X ?T"
        using locally_path_connected_space_open_path_components by blast
      show x: "x ∈ ?T"
        using * nonempty_path_components_of path_component_of_eq path_component_of_eq_empty by fastforce
      have "f ` ?T ⊆ C"
      proof (rule path_components_of_maximal)
        show "C ∈ path_components_of (subtopology Y V)"
          by (simp add: c)
        show "path_connectedin (subtopology Y V) (f ` ?T)"
        proof -
          have "quotient_map (subtopology X {a ∈ topspace X. f a ∈ V}) (subtopology Y V) f"
            by (simp add: V f quotient_map_restriction)
          then show ?thesis
            by (meson path_connectedin_continuous_map_image path_connectedin_path_component_of quotient_imp_continuous_map)
        qed
        show "¬ disjnt C (f ` ?T)"
          by (metis (no_types, lifting) ‹f x ∈ C› x disjnt_iff image_eqI)
      qed
      then show "?T ⊆ {x ∈ topspace X. f x ∈ C}"
        by (force simp: path_component_of_equiv)
    qed
  qed
  then show "openin Y C"
    using f sub by (simp add: quotient_map_def)
qed

lemma homeomorphic_locally_path_connected_space:
  assumes "X homeomorphic_space Y"
  shows "locally_path_connected_space X ⟷ locally_path_connected_space Y"
  using assms
    unfolding homeomorphic_space_def homeomorphic_map_def homeomorphic_maps_map
    by (metis locally_path_connected_space_quotient_map_image)

lemma locally_path_connected_space_retraction_map_image:
   "[retraction_map X Y r; locally_path_connected_space X]
        ==> locally_path_connected_space Y"
  using Abstract_Topology.retraction_imp_quotient_map locally_path_connected_space_quotient_map_image by blast

lemma locally_path_connected_space_euclideanreal: "locally_path_connected_space euclideanreal"
  unfolding locally_path_connected_space_def neighbourhood_base_of
  proof clarsimp
  fix W and x :: "real"
  assume "open W" "x ∈ W"
  then obtain e where "e > 0" and e: "∧x'. ∣x' - x∣ < e ⟶ x' ∈ W"
    by (auto simp: open_real)
  then show "∃U. open U ∧ (∃V. path_connected V ∧ x ∈ U ∧ U ⊆ V ∧ V ⊆ W)"
    by (force intro!: convex_imp_path_connected exI [where x = "{x-e<..])
qed

lemma locally_path_connected_space_discrete_topology:
   "locally_path_connected_space (discrete_topology U)"
  using locally_path_connected_space_open_path_components by fastforce

lemma path_component_eq_connected_component_of:
  assumes "locally_path_connected_space X"
  shows "path_component_of_set X x = connected_component_of_set X x"
proof (cases "x ∈ topspace X")
  case True
  have "path_component_of_set X x ⊆ connected_component_of_set X x"
    by (simp add: path_component_subset_connected_component_of)
  moreover have "closedin X (path_component_of_set X x)"
    by (simp add: assms closedin_path_component_of_locally_path_connected_space)
  moreover have "openin X (path_component_of_set X x)"
    by (simp add: assms openin_path_component_of_locally_path_connected_space)
  moreover have "path_component_of_set X x ≠ {}"
    by (meson True path_component_of_eq_empty)
  ultimately show ?thesis
    using connectedin_connected_component_of [of X x] unfolding connectedin_def
    by (metis closedin_subset_topspace connected_space_clopen_in  
        subset_openin_subtopology topspace_subtopology_subset)
next
  case False
  then show ?thesis
    using connected_component_of_eq_empty path_component_of_eq_empty by fastforce
qed

lemma path_components_eq_connected_components_of:
   "locally_path_connected_space X ==> (path_components_of X = connected_components_of X)"
  by (simp add: path_components_of_def connected_components_of_def image_def path_component_eq_connected_component_of)

lemma path_connected_eq_connected_space:
   "locally_path_connected_space X
         ==> path_connected_space X ⟷ connected_space X"
  by (metis connected_components_of_subset_sing path_components_eq_connected_components_of path_components_of_subset_singleton)

lemma locally_path_connected_space_product_topology:
   "locally_path_connected_space(product_topology X I) ⟷
        (product_topology X I) = trivial_topology ∨
        finite {i. i ∈ I ∧ ~path_connected_space(X i)} ∧
        (∀i ∈ I. locally_path_connected_space(X i))"
    (is "?lhs ⟷ ?empty ∨ ?rhs")
proof (cases ?empty)
  case True
  then show ?thesis
    by (simp add: locally_path_connected_space_def neighbourhood_base_of openin_closedin_eq)
next
  case False
  then obtain z where z: "z ∈ (Π🪙E i∈I. topspace (X i))"
    using discrete_topology_unique_derived_set by (fastforce iff: null_topspace_iff_trivial)
  have ?rhs if L: ?lhs
  proof -
    obtain U C where U: "openin (product_topology X I) U"
      and V: "path_connectedin (product_topology X I) C"
      and "z ∈ U" "U ⊆ C" and Csub: "C ⊆ (Π🪙E i∈I. topspace (X i))"
      by (metis L locally_path_connected_space openin_topspace topspace_product_topology z)
    then obtain V where finV: "finite {i ∈ I. V i ≠ topspace (X i)}"
      and XV: "∧i. i∈I ==> openin (X i) (V i)" and "z ∈ Pi🪙E I V" and subU: "Pi🪙E I V ⊆ U"
      by (force simp: openin_product_topology_alt)
    show ?thesis
    proof (intro conjI ballI)
      have "path_connected_space (X i)" if "i ∈ I" "V i = topspace (X i)" for i
      proof -
        have pc: "path_connectedin (X i) ((λx. x i) ` C)"
          by (metis V continuous_map_product_projection path_connectedin_continuous_map_image that(1))
        moreover have "((λx. x i) ` C) = topspace (X i)"
        proof
          show "(λx. x i) ` C ⊆ topspace (X i)"
            by (simp add: pc path_connectedin_subset_topspace)
          have "V i ⊆ (λx. x i) ` (Π🪙E i∈I. V i)"
            by (metis ‹z ∈ Pi🪙E I V› empty_iff image_projection_PiE order_refl that(1))
          also have "… ⊆ (λx. x i) ` U"
            using subU by blast
          finally show "topspace (X i) ⊆ (λx. x i) ` C"
            using ‹U ⊆ C› that by blast
        qed
        ultimately show ?thesis
          by (simp add: path_connectedin_topspace)
      qed
      then have "{i ∈ I. ¬ path_connected_space (X i)} ⊆ {i ∈ I. V i ≠ topspace (X i)}"
        by blast
      with finV show "finite {i ∈ I. ¬ path_connected_space (X i)}"
        using finite_subset by blast
    next
      show "locally_path_connected_space (X i)" if "i ∈ I" for i
        by (meson False L locally_path_connected_space_quotient_map_image quotient_map_product_projection that)
    qed
  qed
  moreover have ?lhs if R: ?rhs
  proof (clarsimp simp add: locally_path_connected_space_def neighbourhood_base_of)
    fix F z
    assume "openin (product_topology X I) F" and "z ∈ F"
    then obtain W where finW: "finite {i ∈ I. W i ≠ topspace (X i)}"
            and opeW: "∧i. i ∈ I ==> openin (X i) (W i)" and "z ∈ Pi🪙E I W" "Pi🪙E I W ⊆ F"
      by (auto simp: openin_product_topology_alt)
    have "∀i ∈ I. ∃U C. openin (X i) U ∧ path_connectedin (X i) C ∧ z i ∈ U ∧ U ⊆ C ∧ C ⊆ W i ∧
                        (W i = topspace (X i) ∧
                         path_connected_space (X i) ⟶ U = topspace (X i) ∧ C = topspace (X i))"
          (is "∀i ∈ I. ?Φ i")
    proof
      fix i assume "i ∈ I"
      have "locally_path_connected_space (X i)"
        by (simp add: R ‹i ∈ I›)
      moreover have *:"openin (X i) (W i) " "z i ∈ W i"
        using ‹z ∈ Pi🪙E I W› opeW ‹i ∈ I› by auto
      ultimately obtain U C where UC: "openin (X i) U" "path_connectedin (X i) C" "z i ∈ U" "U ⊆ C" "C ⊆ W i"
        using ‹i ∈ I› by (force simp: locally_path_connected_space_def neighbourhood_base_of)
      show "?Φ i"
        by (metis UC * openin_subset path_connectedin_topspace)
    qed
    then obtain U C where
      *: "∧i. i ∈ I ==> openin (X i) (U i) ∧ path_connectedin (X i) (C i) ∧ z i ∈ (U i) ∧ (U i) ⊆ (C i) ∧ (C i) ⊆ W i ∧
                        (W i = topspace (X i) ∧ path_connected_space (X i)
                         ⟶ (U i) = topspace (X i) ∧ (C i) = topspace (X i))"
      by metis
    let ?A = "{i ∈ I. ¬ path_connected_space (X i)} ∪ {i ∈ I. W i ≠ topspace (X i)}"
    have "{i ∈ I. U i ≠ topspace (X i)} ⊆ ?A"
      by (clarsimp simp add: "*")
    moreover have "finite ?A"
      by (simp add: that finW)
    ultimately have "finite {i ∈ I. U i ≠ topspace (X i)}"
      using finite_subset by auto
    with * have "openin (product_topology X I) (Pi🪙E I U)"
      by (simp add: openin_PiE_gen)
    then show "∃U. openin (product_topology X I) U ∧
              (∃V. path_connectedin (product_topology X I) V ∧ z ∈ U ∧ U ⊆ V ∧ V ⊆ F)"
      by (metis (no_types, opaque_lifting) subsetI ‹z ∈ Pi🪙E I W› ‹Pi🪙E I W ⊆ F› * path_connectedin_PiE 
          PiE_iff PiE_mono order.trans)
  qed
  ultimately show ?thesis
    using False by blast
qed

lemma locally_path_connected_is_realinterval:
  assumes "is_interval S"
  shows "locally_path_connected_space(subtopology euclideanreal S)"
  unfolding locally_path_connected_space_def
proof (clarsimp simp add: neighbourhood_base_of openin_subtopology_alt)
  fix a U
  assume "a ∈ S" and "a ∈ U" and "open U"
  then obtain r where "r > 0" and r: "ball a r ⊆ U"
    by (metis open_contains_ball_eq)
  show "∃W. open W ∧ (∃V. path_connectedin (top_of_set S) V ∧ a ∈ W ∧ S ∩ W ⊆ V ∧ V ⊆ S ∧ V ⊆ U)"
  proof (intro exI conjI)
    show "path_connectedin (top_of_set S) (S ∩ ball a r)"
      by (simp add: assms is_interval_Int is_interval_ball_real is_interval_path_connected path_connectedin_subtopology)
    show "a ∈ ball a r"
      by (simp add: ‹0 🚫›)
  qed (use ‹0 🚫› r in auto)
qed

lemma locally_path_connected_real_interval:
 "locally_path_connected_space (subtopology euclideanreal{a..b})"
  "locally_path_connected_space (subtopology euclideanreal{a<..
  using locally_path_connected_is_realinterval
  by (auto simp add: is_interval_1)

lemma locally_path_connected_space_prod_topology:
   "locally_path_connected_space (prod_topology X Y) ⟷
      (prod_topology X Y) = trivial_topology ∨
      locally_path_connected_space X ∧ locally_path_connected_space Y" (is "?lhs=?rhs")
proof (cases "(prod_topology X Y) = trivial_topology")
  case True
  then show ?thesis
    using locally_path_connected_space_discrete_topology by force
next
  case False
  then have ne: "X ≠ trivial_topology" "Y ≠ trivial_topology"
    by simp_all
  show ?thesis
  proof
    assume ?lhs then show ?rhs
      by (meson locally_path_connected_space_quotient_map_image ne(1) ne(2) quotient_map_fst quotient_map_snd)
  next
    assume ?rhs
    with False have X: "locally_path_connected_space X" and Y: "locally_path_connected_space Y"
      by auto
    show ?lhs
      unfolding locally_path_connected_space_def neighbourhood_base_of
    proof clarify
      fix UV x y
      assume UV: "openin (prod_topology X Y) UV" and "(x,y) ∈ UV"
      obtain A B where W12: "openin X A ∧ openin Y B ∧ x ∈ A ∧ y ∈ B ∧ A × B ⊆ UV"
        using X Y by (metis UV ‹(x,y) ∈ UV› openin_prod_topology_alt)
      then obtain C D K L
        where "openin X C" "path_connectedin X K" "x ∈ C" "C ⊆ K" "K ⊆ A"
          "openin Y D" "path_connectedin Y L" "y ∈ D" "D ⊆ L" "L ⊆ B"
        by (metis X Y locally_path_connected_space)
      with W12 ‹openin X C› ‹openin Y D›
      show "∃U V. openin (prod_topology X Y) U ∧ path_connectedin (prod_topology X Y) V ∧ (x, y) ∈ U ∧ U ⊆ V ∧ V ⊆ UV"
        apply (rule_tac x="C × D" in exI)
        apply (rule_tac x="K × L" in exI)
        apply (fastforce simp: openin_prod_Times_iff path_connectedin_Times)
        done
    qed
  qed
qed

lemma locally_path_connected_space_sum_topology:
   "locally_path_connected_space(sum_topology X I) ⟷
    (∀i ∈ I. locally_path_connected_space (X i))" (is "?lhs=?rhs")
proof
  assume ?lhs then show ?rhs
    by (smt (verit) homeomorphic_locally_path_connected_space locally_path_connected_space_open_subset topological_property_of_sum_component)
next
  assume R: ?rhs
  show ?lhs
  proof (clarsimp simp add: locally_path_connected_space_def neighbourhood_base_of forall_openin_sum_topology imp_conjL)
    fix W i x
    assume ope: "∀i∈I. openin (X i) (W i)" 
      and "i ∈ I" and "x ∈ W i"
    then obtain U V where U: "openin (X i) U" and V: "path_connectedin (X i) V" 
           and "x ∈ U" "U ⊆ V" "V ⊆ W i"
      by (metis R ‹i ∈ I› ‹x ∈ W i› locally_path_connected_space)
    show "∃U. openin (sum_topology X I) U ∧ (∃V. path_connectedin (sum_topology X I) V ∧ (i, x) ∈ U ∧ U ⊆ V ∧ V ⊆ Sigma I W)"
    proof (intro exI conjI)
      show "openin (sum_topology X I) (Pair i ` U)"
        by (meson U ‹i ∈ I› open_map_component_injection open_map_def)
      show "path_connectedin (sum_topology X I) (Pair i ` V)"
        by (meson V ‹i ∈ I› continuous_map_component_injection path_connectedin_continuous_map_image)
      show "Pair i ` V ⊆ Sigma I W"
        using ‹V ⊆ W i› ‹i ∈ I› by force
    qed (use ‹x ∈ U› ‹U ⊆ V› in auto)
  qed
qed


subsection‹Locally connected spaces›

definition weakly_locally_connected_at 
  where "weakly_locally_connected_at x X ≡ neighbourhood_base_at x (connectedin X) X"

definition locally_connected_at 
  where "locally_connected_at x X ≡
           neighbourhood_base_at x (λU. openin X U ∧ connectedin X U ) X"

definition locally_connected_space 
  where "locally_connected_space X ≡ neighbourhood_base_of (connectedin X) X"


lemma locally_connected_A: "(∀U x. openin X U ∧ x ∈ U ⟶ openin X (connected_component_of_set (subtopology X U) x))
       ==> neighbourhood_base_of (λU. openin X U ∧ connectedin X U) X"
  unfolding neighbourhood_base_of
  by (metis (full_types) connected_component_of_refl connectedin_connected_component_of connectedin_subtopology mem_Collect_eq openin_subset topspace_subtopology_subset)

lemma locally_connected_B: "locally_connected_space X ==>
          (∀U x. openin X U ∧ x ∈ U ⟶ openin X (connected_component_of_set (subtopology X U) x))"
  unfolding locally_connected_space_def neighbourhood_base_of
  apply (erule all_forward)
  apply clarify
  apply (subst openin_subopen)
  by (smt (verit, ccfv_threshold) Ball_Collect connected_component_of_def connected_component_of_equiv connectedin_subtopology in_mono mem_Collect_eq)

lemma locally_connected_C: "neighbourhood_base_of (λU. openin X U ∧ connectedin X U) X ==> locally_connected_space X"
  using locally_connected_space_def neighbourhood_base_of_mono by auto


lemma locally_connected_space_alt: 
  "locally_connected_space X ⟷ neighbourhood_base_of (λU. openin X U ∧ connectedin X U) X"
  using locally_connected_A locally_connected_B locally_connected_C by blast

lemma locally_connected_space_eq_open_connected_component_of:
  "locally_connected_space X ⟷
        (∀U x. openin X U ∧ x ∈ U
              ⟶ openin X (connected_component_of_set (subtopology X U) x))"
  by (meson locally_connected_A locally_connected_B locally_connected_C)

lemma locally_connected_space:
   "locally_connected_space X ⟷
     (∀V x. openin X V ∧ x ∈ V ⟶ (∃U. openin X U ∧ connectedin X U ∧ x ∈ U ∧ U ⊆ V))"
  by (simp add: locally_connected_space_alt open_neighbourhood_base_of)

lemma locally_path_connected_imp_locally_connected_space:
   "locally_path_connected_space X ==> locally_connected_space X"
  by (simp add: locally_connected_space_def locally_path_connected_space_def neighbourhood_base_of_mono path_connectedin_imp_connectedin)

lemma locally_connected_space_open_connected_components:
  "locally_connected_space X ⟷
   (∀U C. openin X U ∧ C ∈ connected_components_of(subtopology X U) ⟶ openin X C)"
  unfolding connected_components_of_def locally_connected_space_eq_open_connected_component_of
  by (smt (verit, best) image_iff openin_subset topspace_subtopology_subset)

lemma openin_connected_component_of_locally_connected_space:
   "locally_connected_space X ==> openin X (connected_component_of_set X x)"
  by (metis connected_component_of_eq_empty locally_connected_B openin_empty openin_topspace subtopology_topspace)

lemma openin_connected_components_of_locally_connected_space:
   "[locally_connected_space X; C ∈ connected_components_of X] ==> openin X C"
  by (metis locally_connected_space_open_connected_components openin_topspace subtopology_topspace)

lemma weakly_locally_connected_at:
   "weakly_locally_connected_at x X ⟷
    (∀V. openin X V ∧ x ∈ V
       ⟶ (∃U. openin X U ∧ x ∈ U ∧ U ⊆ V ∧
                (∀y ∈ U. ∃C. connectedin X C ∧ C ⊆ V ∧ x ∈ C ∧ y ∈ C)))" (is "?lhs=?rhs")
proof
  assume ?lhs then show ?rhs
    unfolding neighbourhood_base_at_def weakly_locally_connected_at_def
    by (meson subsetD subset_trans)
next
  assume R: ?rhs
  show ?lhs
    unfolding neighbourhood_base_at_def weakly_locally_connected_at_def
  proof clarify
    fix V
    assume "openin X V" and "x ∈ V"
    then obtain U where "openin X U" "x ∈ U" "U ⊆ V" 
                  and U: "∀y∈U. ∃C. connectedin X C ∧ C ⊆ V ∧ x ∈ C ∧ y ∈ C"
      using R by force
    show "∃A B. openin X A ∧ connectedin X B ∧ x ∈ A ∧ A ⊆ B ∧ B ⊆ V"
    proof (intro conjI exI)
      show "connectedin X (connected_component_of_set (subtopology X V) x)"
        by (meson connectedin_connected_component_of connectedin_subtopology)
      show "U ⊆ connected_component_of_set (subtopology X V) x"
        using connected_component_of_maximal U
        by (simp add: connected_component_of_def connectedin_subtopology subsetI)
      show "connected_component_of_set (subtopology X V) x ⊆ V"
        using connected_component_of_subset_topspace by fastforce
    qed (auto simp: ‹x ∈ U› ‹openin X U›)
  qed
qed

lemma locally_connected_space_iff_weak:
  "locally_connected_space X ⟷ (∀x ∈ topspace X. weakly_locally_connected_at x X)"
  by (simp add: locally_connected_space_def neighbourhood_base_of_def weakly_locally_connected_at_def)

lemma locally_connected_space_im_kleinen:
   "locally_connected_space X ⟷
    (∀V x. openin X V ∧ x ∈ V
          ⟶ (∃U. openin X U ∧ x ∈ U ∧ U ⊆ V ∧
                    (∀y ∈ U. ∃C. connectedin X C ∧ C ⊆ V ∧ x ∈ C ∧ y ∈ C)))"
  unfolding locally_connected_space_iff_weak weakly_locally_connected_at
  using openin_subset subsetD by fastforce

lemma locally_connected_space_open_subset:
   "[locally_connected_space X; openin X S] ==> locally_connected_space (subtopology X S)"
  unfolding locally_connected_space_def neighbourhood_base_of
  by (smt (verit) connectedin_subtopology openin_open_subtopology subset_trans)

lemma locally_connected_space_quotient_map_image:
  assumes X: "locally_connected_space X" and f: "quotient_map X Y f"
  shows "locally_connected_space Y"
  unfolding locally_connected_space_open_connected_components
proof clarify
  fix V C
  assume "openin Y V" and C: "C ∈ connected_components_of (subtopology Y V)"
  then have "C ⊆ topspace Y"
    using connected_components_of_subset by force
  have ope1: "openin X {a ∈ topspace X. f a ∈ V}"
    using ‹openin Y V› f openin_continuous_map_preimage quotient_imp_continuous_map by blast
  define Vf where "Vf ≡ {z ∈ topspace X. f z ∈ V}"
  have "openin X {x ∈ topspace X. f x ∈ C}"
  proof (clarsimp simp: openin_subopen [where S = "{x ∈ topspace X. f x ∈ C}"])
    fix x
    assume "x ∈ topspace X" and "f x ∈ C"
    show "∃T. openin X T ∧ x ∈ T ∧ T ⊆ {x ∈ topspace X. f x ∈ C}"
    proof (intro exI conjI)
      show "openin X (connected_component_of_set (subtopology X Vf) x)"
        by (metis Vf_def X connected_component_of_eq_empty locally_connected_B ope1 openin_empty
                  openin_subset topspace_subtopology_subset)
      show x_in_conn: "x ∈ connected_component_of_set (subtopology X Vf) x"
        using C Vf_def ‹f x ∈ C› ‹x ∈ topspace X› connected_component_of_refl connected_components_of_subset by fastforce
      have "connected_component_of_set (subtopology X Vf) x ⊆ topspace X ∩ Vf"
        using connected_component_of_subset_topspace by fastforce
      moreover
      have "f ` connected_component_of_set (subtopology X Vf) x ⊆ C"
      proof (rule connected_components_of_maximal [where X = "subtopology Y V"])
        show "C ∈ connected_components_of (subtopology Y V)"
          by (simp add: C)
        have 🍋: "quotient_map (subtopology X Vf) (subtopology Y V) f"
          by (simp add: Vf_def ‹openin Y V› f quotient_map_restriction)
        then show "connectedin (subtopology Y V) (f ` connected_component_of_set (subtopology X Vf) x)"
          by (metis connectedin_connected_component_of connectedin_continuous_map_image quotient_imp_continuous_map)
        show "¬ disjnt C (f ` connected_component_of_set (subtopology X Vf) x)"
          using ‹f x ∈ C› x_in_conn by (auto simp: disjnt_iff)
      qed
      ultimately
      show "connected_component_of_set (subtopology X Vf) x ⊆ {x ∈ topspace X. f x ∈ C}"
        by blast
    qed
  qed
  then show "openin Y C"
    using ‹C ⊆ topspace Y› f quotient_map_def by fastforce
qed


lemma locally_connected_space_retraction_map_image:
   "[retraction_map X Y r; locally_connected_space X]
        ==> locally_connected_space Y"
  using locally_connected_space_quotient_map_image retraction_imp_quotient_map by blast

lemma homeomorphic_locally_connected_space:
   "X homeomorphic_space Y ==> locally_connected_space X ⟷ locally_connected_space Y"
  by (meson homeomorphic_map_def homeomorphic_space homeomorphic_space_sym locally_connected_space_quotient_map_image)

lemma locally_connected_space_euclideanreal: "locally_connected_space euclideanreal"
  by (simp add: locally_path_connected_imp_locally_connected_space locally_path_connected_space_euclideanreal)

lemma locally_connected_is_realinterval:
   "is_interval S ==> locally_connected_space(subtopology euclideanreal S)"
  by (simp add: locally_path_connected_imp_locally_connected_space locally_path_connected_is_realinterval)

lemma locally_connected_real_interval:
    "locally_connected_space (subtopology euclideanreal{a..b})"
    "locally_connected_space (subtopology euclideanreal{a<..
  using connected_Icc is_interval_connected_1 locally_connected_is_realinterval by auto

lemma locally_connected_space_discrete_topology:
   "locally_connected_space (discrete_topology U)"
  by (simp add: locally_path_connected_imp_locally_connected_space locally_path_connected_space_discrete_topology)

lemma locally_path_connected_imp_locally_connected_at:
   "locally_path_connected_at x X ==> locally_connected_at x X"
  by (simp add: locally_connected_at_def locally_path_connected_at_def neighbourhood_base_at_mono path_connectedin_imp_connectedin)

lemma weakly_locally_path_connected_imp_weakly_locally_connected_at:
   "weakly_locally_path_connected_at x X ==> weakly_locally_connected_at x X"
  by (metis path_connectedin_imp_connectedin weakly_locally_connected_at weakly_locally_path_connected_at)


lemma interior_of_locally_connected_subspace_component:
  assumes X: "locally_connected_space X"
    and C: "C ∈ connected_components_of (subtopology X S)"
  shows "X interior_of C = C ∩ X interior_of S"
proof -
  obtain Csub: "C ⊆ topspace X" "C ⊆ S"
    by (meson C connectedin_connected_components_of connectedin_subset_topspace connectedin_subtopology)
  show ?thesis
  proof
    show "X interior_of C ⊆ C ∩ X interior_of S"
      by (simp add: Csub interior_of_mono interior_of_subset)
    have eq: "X interior_of S = ∪ (connected_components_of (subtopology X (X interior_of S)))"
      by (metis Union_connected_components_of interior_of_subset_topspace topspace_subtopology_subset)
    moreover have "C ∩ D ⊆ X interior_of C"
      if "D ∈ connected_components_of (subtopology X (X interior_of S))" for D
    proof (cases "C ∩ D = {}")
      case False
      have "D ⊆ X interior_of C"
      proof (rule interior_of_maximal)
        have "connectedin (subtopology X S) D"
          by (meson connectedin_connected_components_of connectedin_subtopology interior_of_subset subset_trans that)
        then show "D ⊆ C"
          by (meson C False connected_components_of_maximal disjnt_def)
        show "openin X D"
          using X locally_connected_space_open_connected_components openin_interior_of that by blast
      qed
      then show ?thesis 
        by blast
    qed auto
    ultimately show "C ∩ X interior_of S ⊆ X interior_of C"
      by blast
  qed
qed


lemma frontier_of_locally_connected_subspace_component:
  assumes X: "locally_connected_space X" and "closedin X S" 
    and C: "C ∈ connected_components_of (subtopology X S)"
  shows "X frontier_of C = C ∩ X frontier_of S"
proof -
  obtain Csub: "C ⊆ topspace X" "C ⊆ S"
    by (meson C connectedin_connected_components_of connectedin_subset_topspace connectedin_subtopology)
  then have "X closure_of C - X interior_of C = C ∩ X closure_of S - C ∩ X interior_of S"
    using assms
    apply (simp add: closure_of_closedin flip: interior_of_locally_connected_subspace_component)
    by (metis closedin_connected_components_of closedin_trans_full closure_of_eq inf.orderE)
  then show ?thesis
    by (simp add: Diff_Int_distrib frontier_of_def)
qed

(*Similar proof to locally_connected_space_prod_topology*)
lemma locally_connected_space_prod_topology:
   "locally_connected_space (prod_topology X Y) ⟷
      (prod_topology X Y) = trivial_topology ∨
      locally_connected_space X ∧ locally_connected_space Y" (is "?lhs=?rhs")
proof (cases "(prod_topology X Y) = trivial_topology")
  case True
  then show ?thesis
    using locally_connected_space_iff_weak by force
next
  case False
  then have ne: "X ≠ trivial_topology" "Y ≠ trivial_topology"
    by simp_all
  show ?thesis
  proof
    assume ?lhs then show ?rhs
      by (metis locally_connected_space_quotient_map_image ne quotient_map_fst quotient_map_snd)
  next
    assume ?rhs
    with False have X: "locally_connected_space X" and Y: "locally_connected_space Y"
      by auto
    show ?lhs
      unfolding locally_connected_space_def neighbourhood_base_of
    proof clarify
      fix UV x y
      assume UV: "openin (prod_topology X Y) UV" and "(x,y) ∈ UV"

     obtain A B where W12: "openin X A ∧ openin Y B ∧ x ∈ A ∧ y ∈ B ∧ A × B ⊆ UV"
        using X Y by (metis UV ‹(x,y) ∈ UV› openin_prod_topology_alt)
      then obtain C D K L
        where "openin X C" "connectedin X K" "x ∈ C" "C ⊆ K" "K ⊆ A"
          "openin Y D" "connectedin Y L" "y ∈ D" "D ⊆ L" "L ⊆ B"
        by (metis X Y locally_connected_space)
      with W12 ‹openin X C› ‹openin Y D›
      show "∃U V. openin (prod_topology X Y) U ∧ connectedin (prod_topology X Y) V ∧ (x, y) ∈ U ∧ U ⊆ V ∧ V ⊆ UV"
        apply (rule_tac x="C × D" in exI)
        apply (rule_tac x="K × L" in exI)
        apply (auto simp: openin_prod_Times_iff connectedin_Times)
        done
    qed
  qed
qed

(*Same proof as locally_path_connected_space_product_topology*)
lemma locally_connected_space_product_topology:
   "locally_connected_space(product_topology X I) ⟷
        (product_topology X I) = trivial_topology ∨
        finite {i. i ∈ I ∧ ~connected_space(X i)} ∧
        (∀i ∈ I. locally_connected_space(X i))"
    (is "?lhs ⟷ ?empty ∨ ?rhs")
proof (cases ?empty)
  case True
  then show ?thesis
    by (simp add: locally_connected_space_def neighbourhood_base_of openin_closedin_eq)
next
  case False
  then obtain z where z: "z ∈ (Π🪙E i∈I. topspace (X i))"
    using discrete_topology_unique_derived_set by (fastforce iff: null_topspace_iff_trivial)
  have ?rhs if L: ?lhs
  proof -
    obtain U C where U: "openin (product_topology X I) U"
      and V: "connectedin (product_topology X I) C"
      and "z ∈ U" "U ⊆ C" and Csub: "C ⊆ (Π🪙E i∈I. topspace (X i))"
      using L apply (clarsimp simp add: locally_connected_space_def neighbourhood_base_of)
      by (metis openin_topspace topspace_product_topology z)
    then obtain V where finV: "finite {i ∈ I. V i ≠ topspace (X i)}"
      and XV: "∧i. i∈I ==> openin (X i) (V i)" and "z ∈ Pi🪙E I V" and subU: "Pi🪙E I V ⊆ U"
      by (force simp: openin_product_topology_alt)
    show ?thesis
    proof (intro conjI ballI)
      have "connected_space (X i)" if "i ∈ I" "V i = topspace (X i)" for i
      proof -
        have pc: "connectedin (X i) ((λx. x i) ` C)"
          by (metis V connectedin_continuous_map_image continuous_map_product_projection that(1))
        moreover have "((λx. x i) ` C) = topspace (X i)"
        proof
          show "(λx. x i) ` C ⊆ topspace (X i)"
            by (simp add: pc connectedin_subset_topspace)
          have "V i ⊆ (λx. x i) ` (Π🪙E i∈I. V i)"
            by (metis ‹z ∈ Pi🪙E I V› empty_iff image_projection_PiE order_refl that(1))
          also have "… ⊆ (λx. x i) ` U"
            using subU by blast
          finally show "topspace (X i) ⊆ (λx. x i) ` C"
            using ‹U ⊆ C› that by blast
        qed
        ultimately show ?thesis
          by (simp add: connectedin_topspace)
      qed
      then have "{i ∈ I. ¬ connected_space (X i)} ⊆ {i ∈ I. V i ≠ topspace (X i)}"
        by blast
      with finV show "finite {i ∈ I. ¬ connected_space (X i)}"
        using finite_subset by blast
    next
      show "locally_connected_space (X i)" if "i ∈ I" for i
        by (meson False L locally_connected_space_quotient_map_image quotient_map_product_projection that)
    qed
  qed
  moreover have ?lhs if R: ?rhs
  proof (clarsimp simp add: locally_connected_space_def neighbourhood_base_of)
    fix F z
    assume "openin (product_topology X I) F" and "z ∈ F"
    then obtain W where finW: "finite {i ∈ I. W i ≠ topspace (X i)}"
            and opeW: "∧i. i ∈ I ==> openin (X i) (W i)" and "z ∈ Pi🪙E I W" "Pi🪙E I W ⊆ F"
      by (auto simp: openin_product_topology_alt)
    have "∀i ∈ I. ∃U C. openin (X i) U ∧ connectedin (X i) C ∧ z i ∈ U ∧ U ⊆ C ∧ C ⊆ W i ∧
                        (W i = topspace (X i) ∧
                         connected_space (X i) ⟶ U = topspace (X i) ∧ C = topspace (X i))"
          (is "∀i ∈ I. ?Φ i")
    proof
      fix i assume "i ∈ I"
      have "locally_connected_space (X i)"
        by (simp add: R ‹i ∈ I›)
      moreover have *: "openin (X i) (W i)" "z i ∈ W i"
        using ‹z ∈ Pi🪙E I W› opeW ‹i ∈ I› by auto
      ultimately obtain U C where "openin (X i) U" "connectedin (X i) C" "z i ∈ U" "U ⊆ C" "C ⊆ W i"
        using ‹i ∈ I› by (force simp: locally_connected_space_def neighbourhood_base_of)
      then show "?Φ i"
        by (metis * connectedin_topspace openin_subset)
    qed
    then obtain U C where
      *: "∧i. i ∈ I ==> openin (X i) (U i) ∧ connectedin (X i) (C i) ∧ z i ∈ (U i) ∧ (U i) ⊆ (C i) ∧ (C i) ⊆ W i ∧
                        (W i = topspace (X i) ∧ connected_space (X i)
                         ⟶ (U i) = topspace (X i) ∧ (C i) = topspace (X i))"
      by metis
    let ?A = "{i ∈ I. ¬ connected_space (X i)} ∪ {i ∈ I. W i ≠ topspace (X i)}"
    have "{i ∈ I. U i ≠ topspace (X i)} ⊆ ?A"
      by (clarsimp simp add: "*")
    moreover have "finite ?A"
      by (simp add: that finW)
    ultimately have "finite {i ∈ I. U i ≠ topspace (X i)}"
      using finite_subset by auto
    then have "openin (product_topology X I) (Pi🪙E I U)"
      using * by (simp add: openin_PiE_gen)
    then show "∃U. openin (product_topology X I) U ∧
            (∃V. connectedin (product_topology X I) V ∧ z ∈ U ∧ U ⊆ V ∧ V ⊆ F)"
      using ‹z ∈ Pi🪙E I W› ‹Pi🪙E I W ⊆ F› *
      by (metis (no_types, opaque_lifting) PiE_iff PiE_mono connectedin_PiE subset_iff)
  qed
  ultimately show ?thesis
    using False by blast
qed

lemma locally_connected_space_sum_topology:
   "locally_connected_space(sum_topology X I) ⟷
    (∀i ∈ I. locally_connected_space (X i))" (is "?lhs=?rhs")
proof
  assume ?lhs then show ?rhs
    by (smt (verit) homeomorphic_locally_connected_space locally_connected_space_open_subset topological_property_of_sum_component)
next
  assume R: ?rhs
  show ?lhs
  proof (clarsimp simp add: locally_connected_space_def neighbourhood_base_of forall_openin_sum_topology imp_conjL)
    fix W i x
    assume ope: "∀i∈I. openin (X i) (W i)" 
      and "i ∈ I" and "x ∈ W i"
    then obtain U V where U: "openin (X i) U" and V: "connectedin (X i) V" 
           and "x ∈ U" "U ⊆ V" "V ⊆ W i"
      by (metis R ‹i ∈ I› ‹x ∈ W i› locally_connected_space)
    show "∃U. openin (sum_topology X I) U ∧ (∃V. connectedin (sum_topology X I) V ∧ (i,x) ∈ U ∧ U ⊆ V ∧ V ⊆ Sigma I W)"
    proof (intro exI conjI)
      show "openin (sum_topology X I) (Pair i ` U)"
        by (meson U ‹i ∈ I› open_map_component_injection open_map_def)
      show "connectedin (sum_topology X I) (Pair i ` V)"
        by (meson V ‹i ∈ I› continuous_map_component_injection connectedin_continuous_map_image)
      show "Pair i ` V ⊆ Sigma I W"
        using ‹V ⊆ W i› ‹i ∈ I› by force
    qed (use ‹x ∈ U› ‹U ⊆ V› in auto)
  qed
qed

subsection ‹Dimension of a topological space›

text‹Basic definition of the small inductive dimension relation. Works in any topological space.›

inductive dimension_le :: "['a topology, int] ==> bool" (infix ‹dim'_le› 50) 
  where "[-1 ≤ n;
        ∧V a. [openin X V; a ∈ V] ==> ∃U. a ∈ U ∧ U ⊆ V ∧ openin X U ∧ (subtopology X (X frontier_of U)) dim_le (n-1)]
              ==> X dim_le (n::int)"

lemma dimension_le_neighbourhood_base:
   "X dim_le n ⟷
   -1 ≤ n ∧ neighbourhood_base_of (λU. openin X U ∧ (subtopology X (X frontier_of U)) dim_le (n-1)) X"
  by (smt (verit, best) dimension_le.simps open_neighbourhood_base_of)

lemma dimension_le_bound: "X dim_le n ==>-1 ≤ n"
  using dimension_le.simps by blast
  
lemma dimension_le_mono [rule_format]:
  assumes "X dim_le m"
  shows "m ≤ n ⟶ X dim_le n"
  using assms
proof (induction arbitrary: n rule: dimension_le.induct)
qed (smt (verit) dimension_le.simps)

inductive_simps dim_le_minus2 [simp]: "X dim_le -2"

lemma dimension_le_eq_empty [simp]:
   "X dim_le -1 ⟷ X = trivial_topology"
proof
  show "X dim_le (-1) ==> X = trivial_topology"
    by (force intro: dimension_le.cases)
next
  show "X = trivial_topology ==> X dim_le (-1)"
    using dimension_le.simps openin_subset by fastforce
qed

lemma dimension_le_0_neighbourhood_base_of_clopen:
  "X dim_le 0 ⟷ neighbourhood_base_of (λU. closedin X U ∧ openin X U) X"
proof -
  have "(subtopology X (X frontier_of U) dim_le -1) = closedin X U" 
      if "openin X U" for U
    using that clopenin_eq_frontier_of openin_subset 
    by (fastforce simp add: subtopology_trivial_iff frontier_of_subset_topspace Int_absorb1)
  then show ?thesis
    by (smt (verit, del_insts) dimension_le.simps open_neighbourhood_base_of)
qed

lemma dimension_le_subtopology:
  "X dim_le n ==> subtopology X S dim_le n"
proof (induction arbitrary: S rule: dimension_le.induct)
  case (1 n X)
  show ?case 
  proof (intro dimension_le.intros)
    show "- 1 ≤ n"
      by (simp add: "1.hyps")
    fix U' a
    assume U': "openin (subtopology X S) U'" and "a ∈ U'"
    then obtain U where U: "openin X U" "U' = U ∩ S"
      by (meson openin_subtopology)
    then obtain V where "a ∈ V" "V ⊆ U" "openin X V" 
      and subV: "subtopology X (X frontier_of V) dim_le n-1" 
      and dimV: "∧T. subtopology X (X frontier_of V ∩ T) dim_le n-1"
      by (metis "1.IH" Int_iff ‹a ∈ U'› subtopology_subtopology)
    show "∃W. a ∈ W ∧ W ⊆ U' ∧ openin (subtopology X S) W ∧ subtopology (subtopology X S) (subtopology X S frontier_of W) dim_le n-1"
    proof (intro exI conjI)
      show "a ∈ S ∩ V" "S ∩ V ⊆ U'"
        using ‹U' = U ∩ S› ‹a ∈ U'› ‹a ∈ V› ‹V ⊆ U› by blast+
      show "openin (subtopology X S) (S ∩ V)"
        by (simp add: ‹openin X V› openin_subtopology_Int2)
      have "S ∩ subtopology X S frontier_of V ⊆ X frontier_of V"
        by (simp add: frontier_of_subtopology_subset)
      then show "subtopology (subtopology X S) (subtopology X S frontier_of (S ∩ V)) dim_le n-1"
        by (metis dimV frontier_of_restrict inf.absorb_iff2 inf_left_idem subtopology_subtopology topspace_subtopology)
    qed
  qed
qed

lemma dimension_le_subtopologies:
   "[subtopology X T dim_le n; S ⊆ T] ==> (subtopology X S) dim_le n"
  by (metis dimension_le_subtopology inf.absorb_iff2 subtopology_subtopology)

lemma dimension_le_eq_subtopology:
   "(subtopology X S) dim_le n ⟷
    -1 ≤ n ∧
    (∀V a. openin X V ∧ a ∈ V ∧ a ∈ S
           ⟶ (∃U. a ∈ U ∧ U ⊆ V ∧ openin X U ∧
                    subtopology X (subtopology X S frontier_of (S ∩ U)) dim_le (n-1)))"
proof -
  have *: "(∃T. a ∈ T ∧ T ∩ S ⊆ V ∩ S ∧ openin X T ∧ subtopology X (S ∩ (subtopology X S frontier_of (T ∩ S))) dim_le n-1)
       ⟷ (∃U. a ∈ U ∧ U ⊆ V ∧ openin X U ∧ subtopology X (subtopology X S frontier_of (S ∩ U)) dim_le n-1)"
    if "a ∈ V" "a ∈ S" "openin X V" for a V
  proof -
    have "∃U. a ∈ U ∧ U ⊆ V ∧ openin X U ∧ subtopology X (subtopology X S frontier_of (S ∩ U)) dim_le n-1"
      if "a ∈ T" and sub: "T ∩ S ⊆ V ∩ S" and "openin X T"
        and dim: "subtopology X (S ∩ subtopology X S frontier_of (T ∩ S)) dim_le n-1"
      for T 
    proof (intro exI conjI)
      show "openin X (T ∩ V)"
        using ‹openin X V› ‹openin X T› by blast
      show "subtopology X (subtopology X S frontier_of (S ∩ (T ∩ V))) dim_le n-1"
        by (metis dim frontier_of_subset_subtopology inf.boundedE inf_absorb2 inf_assoc inf_commute sub)
    qed (use ‹a ∈ V› ‹a ∈ T› in auto)
    moreover have "∃T. a ∈ T ∧ T ∩ S ⊆ V ∩ S ∧ openin X T ∧ subtopology X (S ∩ subtopology X S frontier_of (T ∩ S)) dim_le n-1"
      if "a ∈ U" and "U ⊆ V" and "openin X U"
        and dim: "subtopology X (subtopology X S frontier_of (S ∩ U)) dim_le n-1"
      for U
      by (metis that frontier_of_subset_subtopology inf_absorb2 inf_commute inf_le1 le_inf_iff)
    ultimately show ?thesis
      by safe
  qed
  show ?thesis
    apply (simp add: dimension_le.simps [of _ n] subtopology_subtopology openin_subtopology flip: *)
    by (safe; metis Int_iff inf_le2 le_inf_iff)
qed


lemma homeomorphic_space_dimension_le_aux:
  assumes "X homeomorphic_space Y" "X dim_le of_nat n - 1"
  shows "Y dim_le of_nat n - 1"
  using assms
proof (induction n arbitrary: X Y)
  case 0
  then show ?case
    by (simp add: dimension_le_eq_empty homeomorphic_empty_space)
next
  case (Suc n)
  then have X_dim_n: "X dim_le n"
    by simp
  show ?case 
  proof (clarsimp simp add: dimension_le.simps [of Y n])
    fix V b
    assume "openin Y V" and "b ∈ V"
    obtain f g where fg: "homeomorphic_maps X Y f g"
      using ‹X homeomorphic_space Y› homeomorphic_space_def by blast
    then have "openin X (g ` V)"
      using ‹openin Y V› homeomorphic_map_openness_eq homeomorphic_maps_map by blast
    then obtain U where "g b ∈ U" "openin X U" and gim: "U ⊆ g ` V" and sub: "subtopology X (X frontier_of U) dim_le int n - int 1"
      using X_dim_n unfolding dimension_le.simps [of X n] by (metis ‹b ∈ V› imageI of_nat_eq_1_iff)
    show "∃U. b ∈ U ∧ U ⊆ V ∧ openin Y U ∧ subtopology Y (Y frontier_of U) dim_le int n - 1"
    proof (intro conjI exI)
      show "b ∈ f ` U"
        by (metis (no_types, lifting) ‹b ∈ V› ‹g b ∈ U› ‹openin Y V› fg homeomorphic_maps_map image_iff openin_subset subsetD)
      show "f ` U ⊆ V"
        by (smt (verit, ccfv_threshold) ‹openin Y V› fg gim homeomorphic_maps_map image_iff openin_subset subset_iff)
      show "openin Y (f ` U)"
        using ‹openin X U› fg homeomorphic_map_openness_eq homeomorphic_maps_map by blast
      show "subtopology Y (Y frontier_of f ` U) dim_le int n-1"
      proof (rule Suc.IH)
        have "homeomorphic_maps (subtopology X (X frontier_of U)) (subtopology Y (Y frontier_of f ` U)) f g"
          using ‹openin X U› fg
          by (metis frontier_of_subset_topspace homeomorphic_map_frontier_of homeomorphic_maps_map homeomorphic_maps_subtopologies openin_subset topspace_subtopology topspace_subtopology_subset)
        then show "subtopology X (X frontier_of U) homeomorphic_space subtopology Y (Y frontier_of f ` U)"
          using homeomorphic_space_def by blast
        show "subtopology X (X frontier_of U) dim_le int n-1"
          using sub by fastforce
      qed
    qed
  qed
qed

lemma homeomorphic_space_dimension_le:
  assumes "X homeomorphic_space Y"
  shows "X dim_le n ⟷ Y dim_le n"
proof (cases "n ≥ -1")
  case True
  then show ?thesis
    using homeomorphic_space_dimension_le_aux [of _ _ "nat(n+1)"] 
    by (smt (verit) assms homeomorphic_space_sym nat_eq_iff)
next
  case False
  then show ?thesis
    by (metis dimension_le_bound)
qed

lemma dimension_le_retraction_map_image:
   "[retraction_map X Y r; X dim_le n] ==> Y dim_le n"
  by (meson dimension_le_subtopology homeomorphic_space_dimension_le retraction_map_def retraction_maps_section_image2)

lemma dimension_le_discrete_topology [simp]: "(discrete_topology U) dim_le 0"
  using dimension_le.simps dimension_le_eq_empty by fastforce

end