Theory Locally

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  )
         neighbourhood_base_of P X 
             (W x. W    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 "(yU. 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<..<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  (ΠE iI. 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  (ΠE iI. 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. iI  openin (X i) (V i)" and "z  PiE I V" and subU: "PiE 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) ` (ΠE iI. V i)"
            by (metis z  PiE 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  PiE I W" "PiE 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.  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  PiE 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 " 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) (PiE 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  PiE I W PiE 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 < r)
  qed (use 0 < r r in auto)
qed

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

lemma locally_path_connected_space_prod_topology:
   "locally_path_connected_space (prod_topology X Y) 
      (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: "iI. 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: "yU. 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<..<b})"
  using connected_Icc is_interval_connected_1 locally_connected_is_realinterval by auto

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

lemma locally_path_connected_imp_locally_connected_at:
   "locally_path_connected_at x X  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  (ΠE iI. 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  (ΠE iI. 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. iI  openin (X i) (V i)" and "z  PiE I V" and subU: "PiE 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) ` (ΠE iI. V i)"
            by (metis z  PiE 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  PiE I W" "PiE 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.  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  PiE 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 " 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) (PiE 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  PiE I W PiE 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: "iI. 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