Theory Connected

(*  Author:     L C Paulson, University of Cambridge
    Material split off from Topology_Euclidean_Space
*)

section ‹Connected Components›

theory Connected
  imports
    Abstract_Topology_2
begin

subsectiontag unimportant› ‹Connectedness›

lemma connected_local:
 "connected S 
  ¬ (e1 e2.
      openin (top_of_set S) e1 
      openin (top_of_set S) e2 
      S  e1  e2 
      e1  e2 = {} 
      e1  {} 
      e2  {})"
  using connected_openin by blast

lemma exists_diff:
  fixes P :: "'a set  bool"
  shows "(S. P (- S))  (S. P S)"
  by (metis boolean_algebra_class.boolean_algebra.double_compl)

lemma connected_clopen: "connected S 
  (T. openin (top_of_set S) T 
     closedin (top_of_set S) T  T = {}  T = S)" (is "?lhs  ?rhs")
proof -
  have "¬ connected S 
    (e1 e2. open e1  open (- e2)  S  e1  (- e2)  e1  (- e2)  S = {}  e1  S  {}  (- e2)  S  {})"
    unfolding connected_def openin_open closedin_closed
    by (metis double_complement)
  then have th0: "connected S 
    ¬ (e2 e1. closed e2  open e1  S  e1  (- e2)  e1  (- e2)  S = {}  e1  S  {}  (- e2)  S  {})"
    (is " _  ¬ (e2 e1. ?P e2 e1)")
    by (simp add: closed_def) metis
  have th1: "?rhs  ¬ (t' t. closed t't = St'  t{}  tS  (t'. open t'  t = S  t'))"
    (is "_  ¬ (t' t. ?Q t' t)")
    unfolding connected_def openin_open closedin_closed by auto
  have "(e1. ?P e2 e1)  (t. ?Q e2 t)" for e2
  proof -
    have "?P e2 e1  (t. closed e2  t = Se2  open e1  t = Se1  t{}  t  S)" for e1
      by auto
    then show ?thesis
      by metis
  qed
  then have "e2. (e1. ?P e2 e1)  (t. ?Q e2 t)"
    by blast
  then show ?thesis
    by (simp add: th0 th1)
qed

subsection ‹Connected components, considered as a connectedness relation or a set›

definitiontag important› "connected_component S x y  T. connected T  T  S  x  T  y  T"

abbreviation "connected_component_set S x  Collect (connected_component S x)"

lemma connected_componentI:
  "connected T  T  S  x  T  y  T  connected_component S x y"
  by (auto simp: connected_component_def)

lemma connected_component_in: "connected_component S x y  x  S  y  S"
  by (auto simp: connected_component_def)

lemma connected_component_refl: "x  S  connected_component S x x"
  using connected_component_def connected_sing by blast
 
lemma connected_component_refl_eq [simp]: "connected_component S x x  x  S"
  using connected_component_in connected_component_refl by blast

lemma connected_component_sym: "connected_component S x y  connected_component S y x"
  by (auto simp: connected_component_def)

lemma connected_component_trans:
  "connected_component S x y  connected_component S y z  connected_component S x z"
  unfolding connected_component_def
  by (metis Int_iff Un_iff Un_subset_iff equals0D connected_Un)

lemma connected_component_of_subset:
  "connected_component S x y  S  T  connected_component T x y"
  by (auto simp: connected_component_def)

lemma connected_component_Union: "connected_component_set S x = {T. connected T  x  T  T  S}"
  by (auto simp: connected_component_def)

lemma connected_connected_component [iff]: "connected (connected_component_set S x)"
  by (auto simp: connected_component_Union intro: connected_Union)

lemma connected_iff_eq_connected_component_set:
  "connected S  (x  S. connected_component_set S x = S)"
  by (metis connected_component_def connected_component_in connected_connected_component mem_Collect_eq subsetI subset_antisym)

lemma connected_component_subset: "connected_component_set S x  S"
  using connected_component_in by blast

lemma connected_component_eq_self: "connected S  x  S  connected_component_set S x = S"
  by (simp add: connected_iff_eq_connected_component_set)

lemma connected_iff_connected_component:
  "connected S  (x  S. y  S. connected_component S x y)"
  using connected_component_in by (auto simp: connected_iff_eq_connected_component_set)

lemma connected_component_maximal:
  "x  T  connected T  T  S  T  (connected_component_set S x)"
  using connected_component_eq_self connected_component_of_subset by blast

lemma connected_component_mono:
  "S  T  connected_component_set S x  connected_component_set T x"
  by (simp add: Collect_mono connected_component_of_subset)

lemma connected_component_eq_empty [simp]: "connected_component_set S x = {}  x  S"
  using connected_component_refl by (fastforce simp: connected_component_in)

lemma connected_component_set_empty [simp]: "connected_component_set {} x = {}"
  using connected_component_eq_empty by blast

lemma connected_component_eq:
  "y  connected_component_set S x  (connected_component_set S y = connected_component_set S x)"
  by (metis (no_types, lifting)
      Collect_cong connected_component_sym connected_component_trans mem_Collect_eq)

lemma closed_connected_component:
  assumes S: "closed S"
  shows "closed (connected_component_set S x)"
proof (cases "x  S")
  case False
  then show ?thesis
    by (metis connected_component_eq_empty closed_empty)
next
  case True
  show ?thesis
    unfolding closure_eq [symmetric]
  proof
    show "closure (connected_component_set S x)  connected_component_set S x"
    proof (rule connected_component_maximal)
      show "x  closure (connected_component_set S x)"
        by (simp add: closure_def True)
      show "connected (closure (connected_component_set S x))"
        by (simp add: connected_imp_connected_closure)
      show "closure (connected_component_set S x)  S"
        by (simp add: S closure_minimal connected_component_subset)
    qed  
  qed (simp add: closure_subset)
qed

lemma connected_component_disjoint:
  "connected_component_set S a  connected_component_set S b = {} 
    a  connected_component_set S b"
  by (smt (verit) connected_component_eq connected_component_eq_empty connected_component_refl_eq
      disjoint_iff_not_equal mem_Collect_eq)

lemma connected_component_nonoverlap:
  "connected_component_set S a  connected_component_set S b = {} 
    a  S  b  S  connected_component_set S a  connected_component_set S b"
  by (metis connected_component_disjoint connected_component_eq connected_component_eq_empty inf.idem)

lemma connected_component_overlap:
  "connected_component_set S a  connected_component_set S b  {} 
    a  S  b  S  connected_component_set S a = connected_component_set S b"
  by (auto simp: connected_component_nonoverlap)

lemma connected_component_sym_eq: "connected_component S x y  connected_component S y x"
  using connected_component_sym by blast

lemma connected_component_eq_eq:
  "connected_component_set S x = connected_component_set S y 
    x  S  y  S  x  S  y  S  connected_component S x y"
  by (metis connected_component_eq connected_component_eq_empty connected_component_refl mem_Collect_eq)


lemma connected_iff_connected_component_eq:
  "connected S  (x  S. y  S. connected_component_set S x = connected_component_set S y)"
  by (simp add: connected_component_eq_eq connected_iff_connected_component)

lemma connected_component_idemp:
  "connected_component_set (connected_component_set S x) x = connected_component_set S x"
  by (metis Int_absorb connected_component_disjoint connected_component_eq_empty connected_component_eq_self connected_connected_component)

lemma connected_component_unique:
  "x  c; c  S; connected c;
    c'. x  c'; c'  S; connected c'  c'  c
         connected_component_set S x = c"
  by (meson connected_component_maximal connected_component_subset connected_connected_component subsetD subset_antisym)

lemma joinable_connected_component_eq:
  "connected T; T  S;
    connected_component_set S x  T  {};
    connected_component_set S y  T  {}
     connected_component_set S x = connected_component_set S y"
  by (metis (full_types) subsetD connected_component_eq connected_component_maximal disjoint_iff_not_equal)

lemma Union_connected_component: "(connected_component_set S ` S) = S"
proof
  show "(connected_component_set S ` S)  S"
    by (simp add: SUP_least connected_component_subset)
qed (use connected_component_refl_eq in force)

lemma complement_connected_component_unions:
    "S - connected_component_set S x =
     (connected_component_set S ` S - {connected_component_set S x})"
    (is "?lhs = ?rhs")
proof
  show "?lhs  ?rhs"
    by (metis Diff_subset Diff_subset_conv Sup_insert Union_connected_component insert_Diff_single)
  show "?rhs  ?lhs"
    by clarsimp (metis connected_component_eq_eq connected_component_in)
qed

lemma connected_component_intermediate_subset:
        "connected_component_set U a  T; T  U
         connected_component_set T a = connected_component_set U a"
  by (metis connected_component_idemp connected_component_mono subset_antisym)

lemma connected_component_homeomorphismI:
  assumes "homeomorphism A B f g" "connected_component A x y"
  shows   "connected_component B (f x) (f y)"
proof -
  from assms obtain T where T: "connected T" "T  A" "x  T" "y  T"
    unfolding connected_component_def by blast
  have "connected (f ` T)" "f ` T  B" "f x  f ` T" "f y  f ` T"
    using assms T continuous_on_subset[of A f T]
    by (auto intro!: connected_continuous_image simp: homeomorphism_def)
  thus ?thesis
    unfolding connected_component_def by blast
qed

lemma connected_component_homeomorphism_iff:
  assumes "homeomorphism A B f g"
  shows   "connected_component A x y  x  A  y  A  connected_component B (f x) (f y)"
  by (metis assms connected_component_homeomorphismI connected_component_in homeomorphism_apply1 homeomorphism_sym)

lemma connected_component_set_homeomorphism:
  assumes "homeomorphism A B f g" "x  A"
  shows   "connected_component_set B (f x) = f ` connected_component_set A x" (is "?lhs = ?rhs")
proof -
  have "y  ?lhs  y  ?rhs" for y
    by (smt (verit, best) assms connected_component_homeomorphism_iff homeomorphism_def image_iff mem_Collect_eq)
  thus ?thesis
    by blast
qed

subsection ‹The set of connected components of a set›

definitiontag important› components:: "'a::topological_space set  'a set set"
  where "components S  connected_component_set S ` S"

lemma components_iff: "S  components U  (x. x  U  S = connected_component_set U x)"
  by (auto simp: components_def)

lemma componentsI: "x  U  connected_component_set U x  components U"
  by (auto simp: components_def)

lemma componentsE:
  assumes "S  components U"
  obtains x where "x  U" "S = connected_component_set U x"
  using assms by (auto simp: components_def)

lemma Union_components [simp]: "(components U) = U"
  by (simp add: Union_connected_component components_def)

lemma pairwise_disjoint_components: "pairwise (λX Y. X  Y = {}) (components U)"
  unfolding pairwise_def
  by (metis (full_types) components_iff connected_component_nonoverlap)

lemma in_components_nonempty: "C  components S  C  {}"
    by (metis components_iff connected_component_eq_empty)

lemma in_components_subset: "C  components S  C  S"
  using Union_components by blast

lemma in_components_connected: "C  components S  connected C"
  by (metis components_iff connected_connected_component)

lemma in_components_maximal:
  "C  components S 
    C  {}  C  S  connected C  (d. d  {}  C  d  d  S  connected d  d = C)"
(is "?lhs  ?rhs")
proof
  assume L: ?lhs
  then have "C  S" "connected C"
    by (simp_all add: in_components_subset in_components_connected)
  then show ?rhs
    by (metis (full_types) L components_iff connected_component_maximal connected_component_refl empty_iff mem_Collect_eq subsetD subset_antisym)
next
  show "?rhs  ?lhs"
    by (metis bot.extremum_uniqueI components_iff connected_component_eq_empty connected_component_maximal connected_component_subset connected_connected_component subset_emptyI)
qed


lemma joinable_components_eq:
  "connected T  T  S  c1  components S  c2  components S  c1  T  {}  c2  T  {}  c1 = c2"
  by (metis (full_types) components_iff joinable_connected_component_eq)

lemma closed_components: "closed S; C  components S  closed C"
  by (metis closed_connected_component components_iff)

lemma components_nonoverlap:
    "C  components S; C'  components S  (C  C' = {})  (C  C')"
  by (metis (full_types) components_iff connected_component_nonoverlap)

lemma components_eq: "C  components S; C'  components S  (C = C'  C  C'  {})"
  by (metis components_nonoverlap)

lemma components_eq_empty [simp]: "components S = {}  S = {}"
  by (simp add: components_def)

lemma components_empty [simp]: "components {} = {}"
  by simp

lemma connected_eq_connected_components_eq: "connected S  (C  components S. C'  components S. C = C')"
  by (metis (no_types, opaque_lifting) components_iff connected_component_eq_eq connected_iff_connected_component)

lemma components_eq_sing_iff: "components S = {S}  connected S  S  {}" (is "?lhs  ?rhs")
proof
  show "?rhs  ?lhs"
    by (metis components_iff connected_component_eq_self equals0I insert_iff mk_disjoint_insert)
qed (use in_components_connected in fastforce)

lemma components_eq_sing_exists: "(a. components S = {a})  connected S  S  {}"
  by (metis Union_components ccpo_Sup_singleton components_eq_sing_iff)

lemma connected_eq_components_subset_sing: "connected S  components S  {S}"
  by (metis components_eq_empty components_eq_sing_iff connected_empty subset_singleton_iff)

lemma connected_eq_components_subset_sing_exists: "connected S  (a. components S  {a})"
  by (metis components_eq_sing_exists connected_eq_components_subset_sing subset_singleton_iff)

lemma in_components_self: "S  components S  connected S  S  {}"
  by (metis components_empty components_eq_sing_iff empty_iff in_components_connected insertI1)

lemma components_maximal: "C  components S; connected T; T  S; C  T  {}  T  C"
  by (smt (verit, best) Int_Un_eq(4) Un_upper1 bot_eq_sup_iff connected_Un in_components_maximal inf.orderE sup.mono sup.orderI)

lemma exists_component_superset: "T  S; S  {}; connected T  C. C  components S  T  C"
  by (meson componentsI connected_component_maximal equals0I subset_eq)

lemma components_intermediate_subset: "S  components U; S  T; T  U  S  components T"
  by (smt (verit, best) dual_order.trans in_components_maximal)

lemma in_components_unions_complement: "C  components S  S - C = (components S - {C})"
  by (metis complement_connected_component_unions components_def components_iff)

lemma connected_intermediate_closure:
  assumes cs: "connected S" and st: "S  T" and ts: "T  closure S"
  shows "connected T"
  using assms unfolding connected_def
  by (smt (verit) Int_assoc inf.absorb_iff2 inf_bot_left open_Int_closure_eq_empty)

lemma closedin_connected_component: "closedin (top_of_set S) (connected_component_set S x)"
proof (cases "connected_component_set S x = {}")
  case True
  then show ?thesis
    by (metis closedin_empty)
next
  case False
  then obtain y where y: "connected_component S x y" and "x  S"
    using connected_component_eq_empty by blast
  have *: "connected_component_set S x  S  closure (connected_component_set S x)"
    by (auto simp: closure_def connected_component_in)
  have **: "x  closure (connected_component_set S x)"
    by (simp add: x  S closure_def)
  have "S  closure (connected_component_set S x)  connected_component_set S x" if "connected_component S x y"
  proof (rule connected_component_maximal)
    show "connected (S  closure (connected_component_set S x))"
      using "*" connected_intermediate_closure by blast
  qed (use x  S ** in auto)
  with y * show ?thesis
    by (auto simp: closedin_closed)
qed

lemma closedin_component:
   "C  components S  closedin (top_of_set S) C"
  using closedin_connected_component componentsE by blast


subsectiontag unimportant› ‹Proving a function is constant on a connected set
  by proving that a level set is open›

lemma continuous_levelset_openin_cases:
  fixes f :: "_  'b::t1_space"
  shows "connected S  continuous_on S f 
        openin (top_of_set S) {x  S. f x = a}
         (x  S. f x  a)  (x  S. f x = a)"
  unfolding connected_clopen
  using continuous_closedin_preimage_constant by auto

lemma continuous_levelset_openin:
  fixes f :: "_  'b::t1_space"
  shows "connected S  continuous_on S f 
        openin (top_of_set S) {x  S. f x = a} 
        (x  S. f x = a)   (x  S. f x = a)"
  using continuous_levelset_openin_cases[of S f ]
  by meson

lemma continuous_levelset_open:
  fixes f :: "_  'b::t1_space"
  assumes S: "connected S" "continuous_on S f"
    and a: "open {x  S. f x = a}" "x  S.  f x = a"
  shows "x  S. f x = a"
  using a continuous_levelset_openin[OF S, of a, unfolded openin_open]
  by fast


subsectiontag unimportant› ‹Preservation of Connectedness›

lemma homeomorphic_connectedness:
  assumes "S homeomorphic T"
  shows "connected S  connected T"
using assms unfolding homeomorphic_def homeomorphism_def by (metis connected_continuous_image)

lemma connected_monotone_quotient_preimage:
  assumes "connected T"
      and contf: "continuous_on S f" and fim: "f ` S = T"
      and opT: "U. U  T
                  openin (top_of_set S) (S  f -` U) 
                     openin (top_of_set T) U"
      and connT: "y. y  T  connected (S  f -` {y})"
    shows "connected S"
proof (rule connectedI)
  fix U V
  assume "open U" and "open V" and "U  S  {}" and "V  S  {}"
    and "U  V  S = {}" and "S  U  V"
  moreover
  have disjoint: "f ` (S  U)  f ` (S  V) = {}"
  proof -
    have False if "y  f ` (S  U)  f ` (S  V)" for y
    proof -
      have "y  T"
        using fim that by blast
      show ?thesis
        using connectedD [OF connT [OF y  T] open U open V]
              S  U  V U  V  S = {} that by fastforce
    qed
    then show ?thesis by blast
  qed
  ultimately have UU: "(S  f -` f ` (S  U)) = S  U" and VV: "(S  f -` f ` (S  V)) = S  V"
    by auto
  have opeU: "openin (top_of_set T) (f ` (S  U))"
    by (metis UU open U fim image_Int_subset le_inf_iff opT openin_open_Int)
  have opeV: "openin (top_of_set T) (f ` (S  V))"
    by (metis opT fim VV open V openin_open_Int image_Int_subset inf.bounded_iff)
  have "T  f ` (S  U)  f ` (S  V)"
    using S  U  V fim by auto
  then show False
    using connected T disjoint opeU opeV U  S  {} V  S  {}
    by (auto simp: connected_openin)
qed

lemma connected_open_monotone_preimage:
  assumes contf: "continuous_on S f" and fim: "f ` S = T"
    and ST: "C. openin (top_of_set S) C  openin (top_of_set T) (f ` C)"
    and connT: "y. y  T  connected (S  f -` {y})"
    and "connected C" "C  T"
  shows "connected (S  f -` C)"
proof -
  have contf': "continuous_on (S  f -` C) f"
    by (meson contf continuous_on_subset inf_le1)
  have eqC: "f ` (S  f -` C) = C"
    using C  T fim by blast
  show ?thesis
  proof (rule connected_monotone_quotient_preimage [OF connected C contf' eqC])
    show "connected (S  f -` C  f -` {y})" if "y  C" for y
      by (metis Int_assoc Int_empty_right Int_insert_right_if1 assms(6) connT in_mono that vimage_Int)
    have "U. openin (top_of_set (S  f -` C)) U
                openin (top_of_set C) (f ` U)"
      using open_map_restrict [OF _ ST C  T] by metis
    then show "D. D  C
           openin (top_of_set (S  f -` C)) (S  f -` C  f -` D) =
              openin (top_of_set C) D"
      using open_map_imp_quotient_map [of "(S  f -` C)" f] contf' by (simp add: eqC)
  qed
qed


lemma connected_closed_monotone_preimage:
  assumes contf: "continuous_on S f" and fim: "f ` S = T"
    and ST: "C. closedin (top_of_set S) C  closedin (top_of_set T) (f ` C)"
    and connT: "y. y  T  connected (S  f -` {y})"
    and "connected C" "C  T"
  shows "connected (S  f -` C)"
proof -
  have contf': "continuous_on (S  f -` C) f"
    by (meson contf continuous_on_subset inf_le1)
  have eqC: "f ` (S  f -` C) = C"
    using C  T fim by blast
  show ?thesis
  proof (rule connected_monotone_quotient_preimage [OF connected C contf' eqC])
    show "connected (S  f -` C  f -` {y})" if "y  C" for y
      by (metis Int_assoc Int_empty_right Int_insert_right_if1 C  T connT subsetD that vimage_Int)
    have "U. closedin (top_of_set (S  f -` C)) U
                closedin (top_of_set C) (f ` U)"
      using closed_map_restrict [OF _ ST C  T] by metis
    then show "D. D  C
           openin (top_of_set (S  f -` C)) (S  f -` C  f -` D) =
              openin (top_of_set C) D"
      using closed_map_imp_quotient_map [of "(S  f -` C)" f] contf' by (simp add: eqC)
  qed
qed


subsection‹Lemmas about components›

text  ‹See Newman IV, 3.3 and 3.4.›

lemma connected_Un_clopen_in_complement:
  fixes S U :: "'a::metric_space set"
  assumes "connected S" "connected U" "S  U" 
      and opeT: "openin (top_of_set (U - S)) T" 
      and cloT: "closedin (top_of_set (U - S)) T"
    shows "connected (S  T)"
proof -
  have *: "x y. P x y  P y x; x y. P x y  S  x  S  y;
            x y. P x y; S  x  False  ¬(x y. (P x y))" for P
    by metis
  show ?thesis
    unfolding connected_closedin_eq
  proof (rule *)
    fix H1 H2
    assume H: "closedin (top_of_set (S  T)) H1  
               closedin (top_of_set (S  T)) H2 
               H1  H2 = S  T  H1  H2 = {}  H1  {}  H2  {}"
    then have clo: "closedin (top_of_set S) (S  H1)"
                   "closedin (top_of_set S) (S  H2)"
      by (metis Un_upper1 closedin_closed_subset inf_commute)+
    moreover have "S  ((S  T)  H1)  S  ((S  T)  H2) = S"
      using H by blast
    moreover have "H1  (S  ((S  T)  H2)) = {}"
      using H by blast
    ultimately have "S  H1 = {}  S  H2 = {}"
      by (smt (verit) Int_assoc connected S connected_closedin_eq inf_commute inf_sup_absorb)
    then show "S  H1  S  H2"
      using H connected S unfolding connected_closedin by blast
  next
    fix H1 H2
    assume H: "closedin (top_of_set (S  T)) H1 
               closedin (top_of_set (S  T)) H2 
               H1  H2 = S  T  H1  H2 = {}  H1  {}  H2  {}" 
       and "S  H1"
    then have H2T: "H2  T"
      by auto
    have "T  U"
      using Diff_iff opeT openin_imp_subset by auto
    with S  U have Ueq: "U = (U - S)  (S  T)" 
      by auto
    have "openin (top_of_set ((U - S)  (S  T))) H2"
    proof (rule openin_subtopology_Un)
      show "openin (top_of_set (S  T)) H2"
        by (metis Diff_cancel H Un_Diff Un_Diff_Int closedin_subset openin_closedin_eq topspace_euclidean_subtopology)
      then show "openin (top_of_set (U - S)) H2"
        by (meson H2T Un_upper2 opeT openin_subset_trans openin_trans)
    qed
    moreover have "closedin (top_of_set ((U - S)  (S  T))) H2"
    proof (rule closedin_subtopology_Un)
      show "closedin (top_of_set (U - S)) H2"
        using H H2T cloT closedin_subset_trans 
        by (blast intro: closedin_subtopology_Un closedin_trans)
    qed (simp add: H)
    ultimately have H2: "H2 = {}  H2 = U"
      using Ueq connected U unfolding connected_clopen by metis   
    then have "H2  S"
      by (metis Diff_partition H Un_Diff_cancel Un_subset_iff H2  T assms(3) inf.orderE opeT openin_imp_subset)
    moreover have "T  H2 - S"
      by (metis (no_types) H2 H opeT openin_closedin_eq topspace_euclidean_subtopology)
    ultimately show False
      using H S  H1 by blast
  qed blast
qed


proposition component_diff_connected:
  fixes S :: "'a::metric_space set"
  assumes "connected S" "connected U" "S  U" and C: "C  components (U - S)"
  shows "connected(U - C)"
  using connected S unfolding connected_closedin_eq not_ex de_Morgan_conj
proof clarify
  fix H3 H4 
  assume clo3: "closedin (top_of_set (U - C)) H3" 
    and clo4: "closedin (top_of_set (U - C)) H4" 
    and H34: "H3  H4 = U - C" "H3  H4 = {}" and "H3  {}" and "H4  {}"
    and * [rule_format]:
    "H1 H2. ¬ closedin (top_of_set S) H1 
                      ¬ closedin (top_of_set S) H2 
                      H1  H2  S  H1  H2  {}  ¬ H1  {}  ¬ H2  {}"
  then have "H3  U-C" and ope3: "openin (top_of_set (U - C)) (U - C - H3)"
    and "H4  U-C" and ope4: "openin (top_of_set (U - C)) (U - C - H4)"
    by (auto simp: closedin_def)
  have "C  {}" "C  U-S" "connected C"
    using C in_components_nonempty in_components_subset in_components_maximal by blast+
  have cCH3: "connected (C  H3)"
  proof (rule connected_Un_clopen_in_complement [OF connected C connected U _ _ clo3])
    show "openin (top_of_set (U - C)) H3"
      by (metis Diff_cancel Un_Diff Un_Diff_Int H3  H4 = {} H3  H4 = U - C ope4)
  qed (use clo3 C  U - S in auto)
  have cCH4: "connected (C  H4)"
  proof (rule connected_Un_clopen_in_complement [OF connected C connected U _ _ clo4])
    show "openin (top_of_set (U - C)) H4"
      by (metis Diff_cancel Diff_triv Int_Un_eq(2) Un_Diff H34 inf_commute ope3)
  qed (use clo4 C  U - S in auto)
  have "closedin (top_of_set S) (S  H3)" "closedin (top_of_set S) (S  H4)"
    using clo3 clo4 S  U C  U - S by (auto simp: closedin_closed)
  moreover have "S  H3  {}"      
    using components_maximal [OF C cCH3] C  {} C  U - S H3  {} H3  U - C by auto
  moreover have "S  H4  {}"
    using components_maximal [OF C cCH4] C  {} C  U - S H4  {} H4  U - C by auto
  ultimately show False
    using * [of "S  H3" "S  H4"] H3  H4 = {} C  U - S H3  H4 = U - C S  U 
    by auto
qed


subsectiontag unimportant›‹Constancy of a function from a connected set into a finite, disconnected or discrete set›

text‹Still missing: versions for a set that is smaller than R, or countable.›

lemma continuous_disconnected_range_constant:
  assumes S: "connected S"
      and conf: "continuous_on S f"
      and fim: "f  S  T"
      and cct: "y. y  T  connected_component_set T y = {y}"
    shows "f constant_on S"
proof (cases "S = {}")
  case True then show ?thesis
    by (simp add: constant_on_def)
next
  case False
  then have "f ` S  {f x}" if "x  S" for x
    by (metis PiE S cct connected_component_maximal connected_continuous_image [OF conf] fim image_eqI 
        image_subset_iff that)
  with False show ?thesis
    unfolding constant_on_def by blast
qed


text‹This proof requires the existence of two separate values of the range type.›
lemma finite_range_constant_imp_connected:
  assumes "f::'a::topological_space  'b::real_normed_algebra_1.
              continuous_on S f; finite(f ` S)  f constant_on S"
    shows "connected S"
proof -
  { fix T U
    assume clt: "closedin (top_of_set S) T"
       and clu: "closedin (top_of_set S) U"
       and tue: "T  U = {}" and tus: "T  U = S"
    have "continuous_on (T  U) (λx. if x  T then 0 else 1)"
      using clt clu tue by (intro continuous_on_cases_local) (auto simp: tus)
    then have conif: "continuous_on S (λx. if x  T then 0 else 1)"
      using tus by blast
    have fi: "finite ((λx. if x  T then 0 else 1) ` S)"
      by (rule finite_subset [of _ "{0,1}"]) auto
    have "T = {}  U = {}"
      using assms [OF conif fi] tus [symmetric]
      by (auto simp: Ball_def constant_on_def) (metis IntI empty_iff one_neq_zero tue)
  }
  then show ?thesis
    by (simp add: connected_closedin_eq)
qed

end