src/HOL/Analysis/Abstract_Topology_2.thy
author wenzelm
Mon Mar 25 17:21:26 2019 +0100 (3 weeks ago)
changeset 69981 3dced198b9ec
parent 69939 812ce526da33
child 70136 f03a01a18c6e
permissions -rw-r--r--
more strict AFP properties;
     1 (*  Author:     L C Paulson, University of Cambridge
     2     Author:     Amine Chaieb, University of Cambridge
     3     Author:     Robert Himmelmann, TU Muenchen
     4     Author:     Brian Huffman, Portland State University
     5 *)
     6 
     7 section \<open>Abstract Topology 2\<close>
     8 
     9 theory Abstract_Topology_2
    10   imports
    11     Elementary_Topology
    12     Abstract_Topology
    13     "HOL-Library.Indicator_Function"
    14 begin
    15 
    16 text \<open>Combination of Elementary and Abstract Topology\<close>
    17 
    18 (* FIXME: move elsewhere *)
    19 
    20 lemma approachable_lt_le: "(\<exists>(d::real) > 0. \<forall>x. f x < d \<longrightarrow> P x) \<longleftrightarrow> (\<exists>d>0. \<forall>x. f x \<le> d \<longrightarrow> P x)"
    21   apply auto
    22   apply (rule_tac x="d/2" in exI)
    23   apply auto
    24   done
    25 
    26 lemma approachable_lt_le2:  \<comment> \<open>like the above, but pushes aside an extra formula\<close>
    27     "(\<exists>(d::real) > 0. \<forall>x. Q x \<longrightarrow> f x < d \<longrightarrow> P x) \<longleftrightarrow> (\<exists>d>0. \<forall>x. f x \<le> d \<longrightarrow> Q x \<longrightarrow> P x)"
    28   apply auto
    29   apply (rule_tac x="d/2" in exI, auto)
    30   done
    31 
    32 lemma triangle_lemma:
    33   fixes x y z :: real
    34   assumes x: "0 \<le> x"
    35     and y: "0 \<le> y"
    36     and z: "0 \<le> z"
    37     and xy: "x\<^sup>2 \<le> y\<^sup>2 + z\<^sup>2"
    38   shows "x \<le> y + z"
    39 proof -
    40   have "y\<^sup>2 + z\<^sup>2 \<le> y\<^sup>2 + 2 * y * z + z\<^sup>2"
    41     using z y by simp
    42   with xy have th: "x\<^sup>2 \<le> (y + z)\<^sup>2"
    43     by (simp add: power2_eq_square field_simps)
    44   from y z have yz: "y + z \<ge> 0"
    45     by arith
    46   from power2_le_imp_le[OF th yz] show ?thesis .
    47 qed
    48 
    49 lemma isCont_indicator:
    50   fixes x :: "'a::t2_space"
    51   shows "isCont (indicator A :: 'a \<Rightarrow> real) x = (x \<notin> frontier A)"
    52 proof auto
    53   fix x
    54   assume cts_at: "isCont (indicator A :: 'a \<Rightarrow> real) x" and fr: "x \<in> frontier A"
    55   with continuous_at_open have 1: "\<forall>V::real set. open V \<and> indicator A x \<in> V \<longrightarrow>
    56     (\<exists>U::'a set. open U \<and> x \<in> U \<and> (\<forall>y\<in>U. indicator A y \<in> V))" by auto
    57   show False
    58   proof (cases "x \<in> A")
    59     assume x: "x \<in> A"
    60     hence "indicator A x \<in> ({0<..<2} :: real set)" by simp
    61     hence "\<exists>U. open U \<and> x \<in> U \<and> (\<forall>y\<in>U. indicator A y \<in> ({0<..<2} :: real set))"
    62       using 1 open_greaterThanLessThan by blast
    63     then guess U .. note U = this
    64     hence "\<forall>y\<in>U. indicator A y > (0::real)"
    65       unfolding greaterThanLessThan_def by auto
    66     hence "U \<subseteq> A" using indicator_eq_0_iff by force
    67     hence "x \<in> interior A" using U interiorI by auto
    68     thus ?thesis using fr unfolding frontier_def by simp
    69   next
    70     assume x: "x \<notin> A"
    71     hence "indicator A x \<in> ({-1<..<1} :: real set)" by simp
    72     hence "\<exists>U. open U \<and> x \<in> U \<and> (\<forall>y\<in>U. indicator A y \<in> ({-1<..<1} :: real set))"
    73       using 1 open_greaterThanLessThan by blast
    74     then guess U .. note U = this
    75     hence "\<forall>y\<in>U. indicator A y < (1::real)"
    76       unfolding greaterThanLessThan_def by auto
    77     hence "U \<subseteq> -A" by auto
    78     hence "x \<in> interior (-A)" using U interiorI by auto
    79     thus ?thesis using fr interior_complement unfolding frontier_def by auto
    80   qed
    81 next
    82   assume nfr: "x \<notin> frontier A"
    83   hence "x \<in> interior A \<or> x \<in> interior (-A)"
    84     by (auto simp: frontier_def closure_interior)
    85   thus "isCont ((indicator A)::'a \<Rightarrow> real) x"
    86   proof
    87     assume int: "x \<in> interior A"
    88     then obtain U where U: "open U" "x \<in> U" "U \<subseteq> A" unfolding interior_def by auto
    89     hence "\<forall>y\<in>U. indicator A y = (1::real)" unfolding indicator_def by auto
    90     hence "continuous_on U (indicator A)" by (simp add: continuous_on_const indicator_eq_1_iff)
    91     thus ?thesis using U continuous_on_eq_continuous_at by auto
    92   next
    93     assume ext: "x \<in> interior (-A)"
    94     then obtain U where U: "open U" "x \<in> U" "U \<subseteq> -A" unfolding interior_def by auto
    95     then have "continuous_on U (indicator A)"
    96       using continuous_on_topological by (auto simp: subset_iff)
    97     thus ?thesis using U continuous_on_eq_continuous_at by auto
    98   qed
    99 qed
   100 
   101 lemma closedin_limpt:
   102   "closedin (top_of_set T) S \<longleftrightarrow> S \<subseteq> T \<and> (\<forall>x. x islimpt S \<and> x \<in> T \<longrightarrow> x \<in> S)"
   103   apply (simp add: closedin_closed, safe)
   104    apply (simp add: closed_limpt islimpt_subset)
   105   apply (rule_tac x="closure S" in exI, simp)
   106   apply (force simp: closure_def)
   107   done
   108 
   109 lemma closedin_closed_eq: "closed S \<Longrightarrow> closedin (top_of_set S) T \<longleftrightarrow> closed T \<and> T \<subseteq> S"
   110   by (meson closedin_limpt closed_subset closedin_closed_trans)
   111 
   112 lemma connected_closed_set:
   113    "closed S
   114     \<Longrightarrow> connected S \<longleftrightarrow> (\<nexists>A B. closed A \<and> closed B \<and> A \<noteq> {} \<and> B \<noteq> {} \<and> A \<union> B = S \<and> A \<inter> B = {})"
   115   unfolding connected_closedin_eq closedin_closed_eq connected_closedin_eq by blast
   116 
   117 text \<open>If a connnected set is written as the union of two nonempty closed sets, then these sets
   118 have to intersect.\<close>
   119 
   120 lemma connected_as_closed_union:
   121   assumes "connected C" "C = A \<union> B" "closed A" "closed B" "A \<noteq> {}" "B \<noteq> {}"
   122   shows "A \<inter> B \<noteq> {}"
   123 by (metis assms closed_Un connected_closed_set)
   124 
   125 lemma closedin_subset_trans:
   126   "closedin (top_of_set U) S \<Longrightarrow> S \<subseteq> T \<Longrightarrow> T \<subseteq> U \<Longrightarrow>
   127     closedin (top_of_set T) S"
   128   by (meson closedin_limpt subset_iff)
   129 
   130 lemma openin_subset_trans:
   131   "openin (top_of_set U) S \<Longrightarrow> S \<subseteq> T \<Longrightarrow> T \<subseteq> U \<Longrightarrow>
   132     openin (top_of_set T) S"
   133   by (auto simp: openin_open)
   134 
   135 lemma closedin_compact:
   136    "\<lbrakk>compact S; closedin (top_of_set S) T\<rbrakk> \<Longrightarrow> compact T"
   137 by (metis closedin_closed compact_Int_closed)
   138 
   139 lemma closedin_compact_eq:
   140   fixes S :: "'a::t2_space set"
   141   shows
   142    "compact S
   143          \<Longrightarrow> (closedin (top_of_set S) T \<longleftrightarrow>
   144               compact T \<and> T \<subseteq> S)"
   145 by (metis closedin_imp_subset closedin_compact closed_subset compact_imp_closed)
   146 
   147 
   148 subsection \<open>Closure\<close>
   149 
   150 lemma euclidean_closure_of [simp]: "euclidean closure_of S = closure S"
   151   by (auto simp: closure_of_def closure_def islimpt_def)
   152 
   153 lemma closure_openin_Int_closure:
   154   assumes ope: "openin (top_of_set U) S" and "T \<subseteq> U"
   155   shows "closure(S \<inter> closure T) = closure(S \<inter> T)"
   156 proof
   157   obtain V where "open V" and S: "S = U \<inter> V"
   158     using ope using openin_open by metis
   159   show "closure (S \<inter> closure T) \<subseteq> closure (S \<inter> T)"
   160     proof (clarsimp simp: S)
   161       fix x
   162       assume  "x \<in> closure (U \<inter> V \<inter> closure T)"
   163       then have "V \<inter> closure T \<subseteq> A \<Longrightarrow> x \<in> closure A" for A
   164           by (metis closure_mono subsetD inf.coboundedI2 inf_assoc)
   165       then have "x \<in> closure (T \<inter> V)"
   166          by (metis \<open>open V\<close> closure_closure inf_commute open_Int_closure_subset)
   167       then show "x \<in> closure (U \<inter> V \<inter> T)"
   168         by (metis \<open>T \<subseteq> U\<close> inf.absorb_iff2 inf_assoc inf_commute)
   169     qed
   170 next
   171   show "closure (S \<inter> T) \<subseteq> closure (S \<inter> closure T)"
   172     by (meson Int_mono closure_mono closure_subset order_refl)
   173 qed
   174 
   175 corollary infinite_openin:
   176   fixes S :: "'a :: t1_space set"
   177   shows "\<lbrakk>openin (top_of_set U) S; x \<in> S; x islimpt U\<rbrakk> \<Longrightarrow> infinite S"
   178   by (clarsimp simp add: openin_open islimpt_eq_acc_point inf_commute)
   179 
   180 lemma closure_Int_ballI:
   181   assumes "\<And>U. \<lbrakk>openin (top_of_set S) U; U \<noteq> {}\<rbrakk> \<Longrightarrow> T \<inter> U \<noteq> {}"
   182   shows "S \<subseteq> closure T"
   183 proof (clarsimp simp: closure_iff_nhds_not_empty)
   184   fix x and A and V
   185   assume "x \<in> S" "V \<subseteq> A" "open V" "x \<in> V" "T \<inter> A = {}"
   186   then have "openin (top_of_set S) (A \<inter> V \<inter> S)"
   187     by (auto simp: openin_open intro!: exI[where x="V"])
   188   moreover have "A \<inter> V \<inter> S \<noteq> {}" using \<open>x \<in> V\<close> \<open>V \<subseteq> A\<close> \<open>x \<in> S\<close>
   189     by auto
   190   ultimately have "T \<inter> (A \<inter> V \<inter> S) \<noteq> {}"
   191     by (rule assms)
   192   with \<open>T \<inter> A = {}\<close> show False by auto
   193 qed
   194 
   195 
   196 subsection \<open>Frontier\<close>
   197 
   198 lemma euclidean_interior_of [simp]: "euclidean interior_of S = interior S"
   199   by (auto simp: interior_of_def interior_def)
   200 
   201 lemma euclidean_frontier_of [simp]: "euclidean frontier_of S = frontier S"
   202   by (auto simp: frontier_of_def frontier_def)
   203 
   204 lemma connected_Int_frontier:
   205      "\<lbrakk>connected s; s \<inter> t \<noteq> {}; s - t \<noteq> {}\<rbrakk> \<Longrightarrow> (s \<inter> frontier t \<noteq> {})"
   206   apply (simp add: frontier_interiors connected_openin, safe)
   207   apply (drule_tac x="s \<inter> interior t" in spec, safe)
   208    apply (drule_tac [2] x="s \<inter> interior (-t)" in spec)
   209    apply (auto simp: disjoint_eq_subset_Compl dest: interior_subset [THEN subsetD])
   210   done
   211 
   212 subsection \<open>Compactness\<close>
   213 
   214 lemma openin_delete:
   215   fixes a :: "'a :: t1_space"
   216   shows "openin (top_of_set u) s
   217          \<Longrightarrow> openin (top_of_set u) (s - {a})"
   218 by (metis Int_Diff open_delete openin_open)
   219 
   220 lemma compact_eq_openin_cover:
   221   "compact S \<longleftrightarrow>
   222     (\<forall>C. (\<forall>c\<in>C. openin (top_of_set S) c) \<and> S \<subseteq> \<Union>C \<longrightarrow>
   223       (\<exists>D\<subseteq>C. finite D \<and> S \<subseteq> \<Union>D))"
   224 proof safe
   225   fix C
   226   assume "compact S" and "\<forall>c\<in>C. openin (top_of_set S) c" and "S \<subseteq> \<Union>C"
   227   then have "\<forall>c\<in>{T. open T \<and> S \<inter> T \<in> C}. open c" and "S \<subseteq> \<Union>{T. open T \<and> S \<inter> T \<in> C}"
   228     unfolding openin_open by force+
   229   with \<open>compact S\<close> obtain D where "D \<subseteq> {T. open T \<and> S \<inter> T \<in> C}" and "finite D" and "S \<subseteq> \<Union>D"
   230     by (meson compactE)
   231   then have "image (\<lambda>T. S \<inter> T) D \<subseteq> C \<and> finite (image (\<lambda>T. S \<inter> T) D) \<and> S \<subseteq> \<Union>(image (\<lambda>T. S \<inter> T) D)"
   232     by auto
   233   then show "\<exists>D\<subseteq>C. finite D \<and> S \<subseteq> \<Union>D" ..
   234 next
   235   assume 1: "\<forall>C. (\<forall>c\<in>C. openin (top_of_set S) c) \<and> S \<subseteq> \<Union>C \<longrightarrow>
   236         (\<exists>D\<subseteq>C. finite D \<and> S \<subseteq> \<Union>D)"
   237   show "compact S"
   238   proof (rule compactI)
   239     fix C
   240     let ?C = "image (\<lambda>T. S \<inter> T) C"
   241     assume "\<forall>t\<in>C. open t" and "S \<subseteq> \<Union>C"
   242     then have "(\<forall>c\<in>?C. openin (top_of_set S) c) \<and> S \<subseteq> \<Union>?C"
   243       unfolding openin_open by auto
   244     with 1 obtain D where "D \<subseteq> ?C" and "finite D" and "S \<subseteq> \<Union>D"
   245       by metis
   246     let ?D = "inv_into C (\<lambda>T. S \<inter> T) ` D"
   247     have "?D \<subseteq> C \<and> finite ?D \<and> S \<subseteq> \<Union>?D"
   248     proof (intro conjI)
   249       from \<open>D \<subseteq> ?C\<close> show "?D \<subseteq> C"
   250         by (fast intro: inv_into_into)
   251       from \<open>finite D\<close> show "finite ?D"
   252         by (rule finite_imageI)
   253       from \<open>S \<subseteq> \<Union>D\<close> show "S \<subseteq> \<Union>?D"
   254         apply (rule subset_trans, clarsimp)
   255         apply (frule subsetD [OF \<open>D \<subseteq> ?C\<close>, THEN f_inv_into_f])
   256         apply (erule rev_bexI, fast)
   257         done
   258     qed
   259     then show "\<exists>D\<subseteq>C. finite D \<and> S \<subseteq> \<Union>D" ..
   260   qed
   261 qed
   262 
   263 
   264 subsection \<open>Continuity\<close>
   265 
   266 lemma interior_image_subset:
   267   assumes "inj f" "\<And>x. continuous (at x) f"
   268   shows "interior (f ` S) \<subseteq> f ` (interior S)"
   269 proof
   270   fix x assume "x \<in> interior (f ` S)"
   271   then obtain T where as: "open T" "x \<in> T" "T \<subseteq> f ` S" ..
   272   then have "x \<in> f ` S" by auto
   273   then obtain y where y: "y \<in> S" "x = f y" by auto
   274   have "open (f -` T)"
   275     using assms \<open>open T\<close> by (simp add: continuous_at_imp_continuous_on open_vimage)
   276   moreover have "y \<in> vimage f T"
   277     using \<open>x = f y\<close> \<open>x \<in> T\<close> by simp
   278   moreover have "vimage f T \<subseteq> S"
   279     using \<open>T \<subseteq> image f S\<close> \<open>inj f\<close> unfolding inj_on_def subset_eq by auto
   280   ultimately have "y \<in> interior S" ..
   281   with \<open>x = f y\<close> show "x \<in> f ` interior S" ..
   282 qed
   283 
   284 subsection%unimportant \<open>Equality of continuous functions on closure and related results\<close>
   285 
   286 lemma continuous_closedin_preimage_constant:
   287   fixes f :: "_ \<Rightarrow> 'b::t1_space"
   288   shows "continuous_on S f \<Longrightarrow> closedin (top_of_set S) {x \<in> S. f x = a}"
   289   using continuous_closedin_preimage[of S f "{a}"] by (simp add: vimage_def Collect_conj_eq)
   290 
   291 lemma continuous_closed_preimage_constant:
   292   fixes f :: "_ \<Rightarrow> 'b::t1_space"
   293   shows "continuous_on S f \<Longrightarrow> closed S \<Longrightarrow> closed {x \<in> S. f x = a}"
   294   using continuous_closed_preimage[of S f "{a}"] by (simp add: vimage_def Collect_conj_eq)
   295 
   296 lemma continuous_constant_on_closure:
   297   fixes f :: "_ \<Rightarrow> 'b::t1_space"
   298   assumes "continuous_on (closure S) f"
   299       and "\<And>x. x \<in> S \<Longrightarrow> f x = a"
   300       and "x \<in> closure S"
   301   shows "f x = a"
   302     using continuous_closed_preimage_constant[of "closure S" f a]
   303       assms closure_minimal[of S "{x \<in> closure S. f x = a}"] closure_subset
   304     unfolding subset_eq
   305     by auto
   306 
   307 lemma image_closure_subset:
   308   assumes contf: "continuous_on (closure S) f"
   309     and "closed T"
   310     and "(f ` S) \<subseteq> T"
   311   shows "f ` (closure S) \<subseteq> T"
   312 proof -
   313   have "S \<subseteq> {x \<in> closure S. f x \<in> T}"
   314     using assms(3) closure_subset by auto
   315   moreover have "closed (closure S \<inter> f -` T)"
   316     using continuous_closed_preimage[OF contf] \<open>closed T\<close> by auto
   317   ultimately have "closure S = (closure S \<inter> f -` T)"
   318     using closure_minimal[of S "(closure S \<inter> f -` T)"] by auto
   319   then show ?thesis by auto
   320 qed
   321 
   322 subsection%unimportant \<open>A function constant on a set\<close>
   323 
   324 definition constant_on  (infixl "(constant'_on)" 50)
   325   where "f constant_on A \<equiv> \<exists>y. \<forall>x\<in>A. f x = y"
   326 
   327 lemma constant_on_subset: "\<lbrakk>f constant_on A; B \<subseteq> A\<rbrakk> \<Longrightarrow> f constant_on B"
   328   unfolding constant_on_def by blast
   329 
   330 lemma injective_not_constant:
   331   fixes S :: "'a::{perfect_space} set"
   332   shows "\<lbrakk>open S; inj_on f S; f constant_on S\<rbrakk> \<Longrightarrow> S = {}"
   333 unfolding constant_on_def
   334 by (metis equals0I inj_on_contraD islimpt_UNIV islimpt_def)
   335 
   336 lemma constant_on_closureI:
   337   fixes f :: "_ \<Rightarrow> 'b::t1_space"
   338   assumes cof: "f constant_on S" and contf: "continuous_on (closure S) f"
   339     shows "f constant_on (closure S)"
   340 using continuous_constant_on_closure [OF contf] cof unfolding constant_on_def
   341 by metis
   342 
   343 
   344 subsection%unimportant \<open>Continuity relative to a union.\<close>
   345 
   346 lemma continuous_on_Un_local:
   347     "\<lbrakk>closedin (top_of_set (s \<union> t)) s; closedin (top_of_set (s \<union> t)) t;
   348       continuous_on s f; continuous_on t f\<rbrakk>
   349      \<Longrightarrow> continuous_on (s \<union> t) f"
   350   unfolding continuous_on closedin_limpt
   351   by (metis Lim_trivial_limit Lim_within_union Un_iff trivial_limit_within)
   352 
   353 lemma continuous_on_cases_local:
   354      "\<lbrakk>closedin (top_of_set (s \<union> t)) s; closedin (top_of_set (s \<union> t)) t;
   355        continuous_on s f; continuous_on t g;
   356        \<And>x. \<lbrakk>x \<in> s \<and> \<not>P x \<or> x \<in> t \<and> P x\<rbrakk> \<Longrightarrow> f x = g x\<rbrakk>
   357       \<Longrightarrow> continuous_on (s \<union> t) (\<lambda>x. if P x then f x else g x)"
   358   by (rule continuous_on_Un_local) (auto intro: continuous_on_eq)
   359 
   360 lemma continuous_on_cases_le:
   361   fixes h :: "'a :: topological_space \<Rightarrow> real"
   362   assumes "continuous_on {t \<in> s. h t \<le> a} f"
   363       and "continuous_on {t \<in> s. a \<le> h t} g"
   364       and h: "continuous_on s h"
   365       and "\<And>t. \<lbrakk>t \<in> s; h t = a\<rbrakk> \<Longrightarrow> f t = g t"
   366     shows "continuous_on s (\<lambda>t. if h t \<le> a then f(t) else g(t))"
   367 proof -
   368   have s: "s = (s \<inter> h -` atMost a) \<union> (s \<inter> h -` atLeast a)"
   369     by force
   370   have 1: "closedin (top_of_set s) (s \<inter> h -` atMost a)"
   371     by (rule continuous_closedin_preimage [OF h closed_atMost])
   372   have 2: "closedin (top_of_set s) (s \<inter> h -` atLeast a)"
   373     by (rule continuous_closedin_preimage [OF h closed_atLeast])
   374   have eq: "s \<inter> h -` {..a} = {t \<in> s. h t \<le> a}" "s \<inter> h -` {a..} = {t \<in> s. a \<le> h t}"
   375     by auto
   376   show ?thesis
   377     apply (rule continuous_on_subset [of s, OF _ order_refl])
   378     apply (subst s)
   379     apply (rule continuous_on_cases_local)
   380     using 1 2 s assms apply (auto simp: eq)
   381     done
   382 qed
   383 
   384 lemma continuous_on_cases_1:
   385   fixes s :: "real set"
   386   assumes "continuous_on {t \<in> s. t \<le> a} f"
   387       and "continuous_on {t \<in> s. a \<le> t} g"
   388       and "a \<in> s \<Longrightarrow> f a = g a"
   389     shows "continuous_on s (\<lambda>t. if t \<le> a then f(t) else g(t))"
   390 using assms
   391 by (auto simp: continuous_on_id intro: continuous_on_cases_le [where h = id, simplified])
   392 
   393 
   394 subsection%unimportant\<open>Inverse function property for open/closed maps\<close>
   395 
   396 lemma continuous_on_inverse_open_map:
   397   assumes contf: "continuous_on S f"
   398     and imf: "f ` S = T"
   399     and injf: "\<And>x. x \<in> S \<Longrightarrow> g (f x) = x"
   400     and oo: "\<And>U. openin (top_of_set S) U \<Longrightarrow> openin (top_of_set T) (f ` U)"
   401   shows "continuous_on T g"
   402 proof -
   403   from imf injf have gTS: "g ` T = S"
   404     by force
   405   from imf injf have fU: "U \<subseteq> S \<Longrightarrow> (f ` U) = T \<inter> g -` U" for U
   406     by force
   407   show ?thesis
   408     by (simp add: continuous_on_open [of T g] gTS) (metis openin_imp_subset fU oo)
   409 qed
   410 
   411 lemma continuous_on_inverse_closed_map:
   412   assumes contf: "continuous_on S f"
   413     and imf: "f ` S = T"
   414     and injf: "\<And>x. x \<in> S \<Longrightarrow> g(f x) = x"
   415     and oo: "\<And>U. closedin (top_of_set S) U \<Longrightarrow> closedin (top_of_set T) (f ` U)"
   416   shows "continuous_on T g"
   417 proof -
   418   from imf injf have gTS: "g ` T = S"
   419     by force
   420   from imf injf have fU: "U \<subseteq> S \<Longrightarrow> (f ` U) = T \<inter> g -` U" for U
   421     by force
   422   show ?thesis
   423     by (simp add: continuous_on_closed [of T g] gTS) (metis closedin_imp_subset fU oo)
   424 qed
   425 
   426 lemma homeomorphism_injective_open_map:
   427   assumes contf: "continuous_on S f"
   428     and imf: "f ` S = T"
   429     and injf: "inj_on f S"
   430     and oo: "\<And>U. openin (top_of_set S) U \<Longrightarrow> openin (top_of_set T) (f ` U)"
   431   obtains g where "homeomorphism S T f g"
   432 proof
   433   have "continuous_on T (inv_into S f)"
   434     by (metis contf continuous_on_inverse_open_map imf injf inv_into_f_f oo)
   435   with imf injf contf show "homeomorphism S T f (inv_into S f)"
   436     by (auto simp: homeomorphism_def)
   437 qed
   438 
   439 lemma homeomorphism_injective_closed_map:
   440   assumes contf: "continuous_on S f"
   441     and imf: "f ` S = T"
   442     and injf: "inj_on f S"
   443     and oo: "\<And>U. closedin (top_of_set S) U \<Longrightarrow> closedin (top_of_set T) (f ` U)"
   444   obtains g where "homeomorphism S T f g"
   445 proof
   446   have "continuous_on T (inv_into S f)"
   447     by (metis contf continuous_on_inverse_closed_map imf injf inv_into_f_f oo)
   448   with imf injf contf show "homeomorphism S T f (inv_into S f)"
   449     by (auto simp: homeomorphism_def)
   450 qed
   451 
   452 lemma homeomorphism_imp_open_map:
   453   assumes hom: "homeomorphism S T f g"
   454     and oo: "openin (top_of_set S) U"
   455   shows "openin (top_of_set T) (f ` U)"
   456 proof -
   457   from hom oo have [simp]: "f ` U = T \<inter> g -` U"
   458     using openin_subset by (fastforce simp: homeomorphism_def rev_image_eqI)
   459   from hom have "continuous_on T g"
   460     unfolding homeomorphism_def by blast
   461   moreover have "g ` T = S"
   462     by (metis hom homeomorphism_def)
   463   ultimately show ?thesis
   464     by (simp add: continuous_on_open oo)
   465 qed
   466 
   467 lemma homeomorphism_imp_closed_map:
   468   assumes hom: "homeomorphism S T f g"
   469     and oo: "closedin (top_of_set S) U"
   470   shows "closedin (top_of_set T) (f ` U)"
   471 proof -
   472   from hom oo have [simp]: "f ` U = T \<inter> g -` U"
   473     using closedin_subset by (fastforce simp: homeomorphism_def rev_image_eqI)
   474   from hom have "continuous_on T g"
   475     unfolding homeomorphism_def by blast
   476   moreover have "g ` T = S"
   477     by (metis hom homeomorphism_def)
   478   ultimately show ?thesis
   479     by (simp add: continuous_on_closed oo)
   480 qed
   481 
   482 subsection%unimportant \<open>Seperability\<close>
   483 
   484 lemma subset_second_countable:
   485   obtains \<B> :: "'a:: second_countable_topology set set"
   486     where "countable \<B>"
   487           "{} \<notin> \<B>"
   488           "\<And>C. C \<in> \<B> \<Longrightarrow> openin(top_of_set S) C"
   489           "\<And>T. openin(top_of_set S) T \<Longrightarrow> \<exists>\<U>. \<U> \<subseteq> \<B> \<and> T = \<Union>\<U>"
   490 proof -
   491   obtain \<B> :: "'a set set"
   492     where "countable \<B>"
   493       and opeB: "\<And>C. C \<in> \<B> \<Longrightarrow> openin(top_of_set S) C"
   494       and \<B>:    "\<And>T. openin(top_of_set S) T \<Longrightarrow> \<exists>\<U>. \<U> \<subseteq> \<B> \<and> T = \<Union>\<U>"
   495   proof -
   496     obtain \<C> :: "'a set set"
   497       where "countable \<C>" and ope: "\<And>C. C \<in> \<C> \<Longrightarrow> open C"
   498         and \<C>: "\<And>S. open S \<Longrightarrow> \<exists>U. U \<subseteq> \<C> \<and> S = \<Union>U"
   499       by (metis univ_second_countable that)
   500     show ?thesis
   501     proof
   502       show "countable ((\<lambda>C. S \<inter> C) ` \<C>)"
   503         by (simp add: \<open>countable \<C>\<close>)
   504       show "\<And>C. C \<in> (\<inter>) S ` \<C> \<Longrightarrow> openin (top_of_set S) C"
   505         using ope by auto
   506       show "\<And>T. openin (top_of_set S) T \<Longrightarrow> \<exists>\<U>\<subseteq>(\<inter>) S ` \<C>. T = \<Union>\<U>"
   507         by (metis \<C> image_mono inf_Sup openin_open)
   508     qed
   509   qed
   510   show ?thesis
   511   proof
   512     show "countable (\<B> - {{}})"
   513       using \<open>countable \<B>\<close> by blast
   514     show "\<And>C. \<lbrakk>C \<in> \<B> - {{}}\<rbrakk> \<Longrightarrow> openin (top_of_set S) C"
   515       by (simp add: \<open>\<And>C. C \<in> \<B> \<Longrightarrow> openin (top_of_set S) C\<close>)
   516     show "\<exists>\<U>\<subseteq>\<B> - {{}}. T = \<Union>\<U>" if "openin (top_of_set S) T" for T
   517       using \<B> [OF that]
   518       apply clarify
   519       apply (rule_tac x="\<U> - {{}}" in exI, auto)
   520         done
   521   qed auto
   522 qed
   523 
   524 lemma Lindelof_openin:
   525   fixes \<F> :: "'a::second_countable_topology set set"
   526   assumes "\<And>S. S \<in> \<F> \<Longrightarrow> openin (top_of_set U) S"
   527   obtains \<F>' where "\<F>' \<subseteq> \<F>" "countable \<F>'" "\<Union>\<F>' = \<Union>\<F>"
   528 proof -
   529   have "\<And>S. S \<in> \<F> \<Longrightarrow> \<exists>T. open T \<and> S = U \<inter> T"
   530     using assms by (simp add: openin_open)
   531   then obtain tf where tf: "\<And>S. S \<in> \<F> \<Longrightarrow> open (tf S) \<and> (S = U \<inter> tf S)"
   532     by metis
   533   have [simp]: "\<And>\<F>'. \<F>' \<subseteq> \<F> \<Longrightarrow> \<Union>\<F>' = U \<inter> \<Union>(tf ` \<F>')"
   534     using tf by fastforce
   535   obtain \<G> where "countable \<G> \<and> \<G> \<subseteq> tf ` \<F>" "\<Union>\<G> = \<Union>(tf ` \<F>)"
   536     using tf by (force intro: Lindelof [of "tf ` \<F>"])
   537   then obtain \<F>' where \<F>': "\<F>' \<subseteq> \<F>" "countable \<F>'" "\<Union>\<F>' = \<Union>\<F>"
   538     by (clarsimp simp add: countable_subset_image)
   539   then show ?thesis ..
   540 qed
   541 
   542 
   543 subsection%unimportant\<open>Closed Maps\<close>
   544 
   545 lemma continuous_imp_closed_map:
   546   fixes f :: "'a::t2_space \<Rightarrow> 'b::t2_space"
   547   assumes "closedin (top_of_set S) U"
   548           "continuous_on S f" "f ` S = T" "compact S"
   549     shows "closedin (top_of_set T) (f ` U)"
   550   by (metis assms closedin_compact_eq compact_continuous_image continuous_on_subset subset_image_iff)
   551 
   552 lemma closed_map_restrict:
   553   assumes cloU: "closedin (top_of_set (S \<inter> f -` T')) U"
   554     and cc: "\<And>U. closedin (top_of_set S) U \<Longrightarrow> closedin (top_of_set T) (f ` U)"
   555     and "T' \<subseteq> T"
   556   shows "closedin (top_of_set T') (f ` U)"
   557 proof -
   558   obtain V where "closed V" "U = S \<inter> f -` T' \<inter> V"
   559     using cloU by (auto simp: closedin_closed)
   560   with cc [of "S \<inter> V"] \<open>T' \<subseteq> T\<close> show ?thesis
   561     by (fastforce simp add: closedin_closed)
   562 qed
   563 
   564 subsection%unimportant\<open>Open Maps\<close>
   565 
   566 lemma open_map_restrict:
   567   assumes opeU: "openin (top_of_set (S \<inter> f -` T')) U"
   568     and oo: "\<And>U. openin (top_of_set S) U \<Longrightarrow> openin (top_of_set T) (f ` U)"
   569     and "T' \<subseteq> T"
   570   shows "openin (top_of_set T') (f ` U)"
   571 proof -
   572   obtain V where "open V" "U = S \<inter> f -` T' \<inter> V"
   573     using opeU by (auto simp: openin_open)
   574   with oo [of "S \<inter> V"] \<open>T' \<subseteq> T\<close> show ?thesis
   575     by (fastforce simp add: openin_open)
   576 qed
   577 
   578 
   579 subsection%unimportant\<open>Quotient maps\<close>
   580 
   581 lemma quotient_map_imp_continuous_open:
   582   assumes T: "f ` S \<subseteq> T"
   583       and ope: "\<And>U. U \<subseteq> T
   584               \<Longrightarrow> (openin (top_of_set S) (S \<inter> f -` U) \<longleftrightarrow>
   585                    openin (top_of_set T) U)"
   586     shows "continuous_on S f"
   587 proof -
   588   have [simp]: "S \<inter> f -` f ` S = S" by auto
   589   show ?thesis
   590     using ope [OF T]
   591     apply (simp add: continuous_on_open)
   592     by (meson ope openin_imp_subset openin_trans)
   593 qed
   594 
   595 lemma quotient_map_imp_continuous_closed:
   596   assumes T: "f ` S \<subseteq> T"
   597       and ope: "\<And>U. U \<subseteq> T
   598                   \<Longrightarrow> (closedin (top_of_set S) (S \<inter> f -` U) \<longleftrightarrow>
   599                        closedin (top_of_set T) U)"
   600     shows "continuous_on S f"
   601 proof -
   602   have [simp]: "S \<inter> f -` f ` S = S" by auto
   603   show ?thesis
   604     using ope [OF T]
   605     apply (simp add: continuous_on_closed)
   606     by (metis (no_types, lifting) ope closedin_imp_subset closedin_trans)
   607 qed
   608 
   609 lemma open_map_imp_quotient_map:
   610   assumes contf: "continuous_on S f"
   611       and T: "T \<subseteq> f ` S"
   612       and ope: "\<And>T. openin (top_of_set S) T
   613                    \<Longrightarrow> openin (top_of_set (f ` S)) (f ` T)"
   614     shows "openin (top_of_set S) (S \<inter> f -` T) =
   615            openin (top_of_set (f ` S)) T"
   616 proof -
   617   have "T = f ` (S \<inter> f -` T)"
   618     using T by blast
   619   then show ?thesis
   620     using "ope" contf continuous_on_open by metis
   621 qed
   622 
   623 lemma closed_map_imp_quotient_map:
   624   assumes contf: "continuous_on S f"
   625       and T: "T \<subseteq> f ` S"
   626       and ope: "\<And>T. closedin (top_of_set S) T
   627               \<Longrightarrow> closedin (top_of_set (f ` S)) (f ` T)"
   628     shows "openin (top_of_set S) (S \<inter> f -` T) \<longleftrightarrow>
   629            openin (top_of_set (f ` S)) T"
   630           (is "?lhs = ?rhs")
   631 proof
   632   assume ?lhs
   633   then have *: "closedin (top_of_set S) (S - (S \<inter> f -` T))"
   634     using closedin_diff by fastforce
   635   have [simp]: "(f ` S - f ` (S - (S \<inter> f -` T))) = T"
   636     using T by blast
   637   show ?rhs
   638     using ope [OF *, unfolded closedin_def] by auto
   639 next
   640   assume ?rhs
   641   with contf show ?lhs
   642     by (auto simp: continuous_on_open)
   643 qed
   644 
   645 lemma continuous_right_inverse_imp_quotient_map:
   646   assumes contf: "continuous_on S f" and imf: "f ` S \<subseteq> T"
   647       and contg: "continuous_on T g" and img: "g ` T \<subseteq> S"
   648       and fg [simp]: "\<And>y. y \<in> T \<Longrightarrow> f(g y) = y"
   649       and U: "U \<subseteq> T"
   650     shows "openin (top_of_set S) (S \<inter> f -` U) \<longleftrightarrow>
   651            openin (top_of_set T) U"
   652           (is "?lhs = ?rhs")
   653 proof -
   654   have f: "\<And>Z. openin (top_of_set (f ` S)) Z \<Longrightarrow>
   655                 openin (top_of_set S) (S \<inter> f -` Z)"
   656   and  g: "\<And>Z. openin (top_of_set (g ` T)) Z \<Longrightarrow>
   657                 openin (top_of_set T) (T \<inter> g -` Z)"
   658     using contf contg by (auto simp: continuous_on_open)
   659   show ?thesis
   660   proof
   661     have "T \<inter> g -` (g ` T \<inter> (S \<inter> f -` U)) = {x \<in> T. f (g x) \<in> U}"
   662       using imf img by blast
   663     also have "... = U"
   664       using U by auto
   665     finally have eq: "T \<inter> g -` (g ` T \<inter> (S \<inter> f -` U)) = U" .
   666     assume ?lhs
   667     then have *: "openin (top_of_set (g ` T)) (g ` T \<inter> (S \<inter> f -` U))"
   668       by (meson img openin_Int openin_subtopology_Int_subset openin_subtopology_self)
   669     show ?rhs
   670       using g [OF *] eq by auto
   671   next
   672     assume rhs: ?rhs
   673     show ?lhs
   674       by (metis f fg image_eqI image_subset_iff imf img openin_subopen openin_subtopology_self openin_trans rhs)
   675   qed
   676 qed
   677 
   678 lemma continuous_left_inverse_imp_quotient_map:
   679   assumes "continuous_on S f"
   680       and "continuous_on (f ` S) g"
   681       and  "\<And>x. x \<in> S \<Longrightarrow> g(f x) = x"
   682       and "U \<subseteq> f ` S"
   683     shows "openin (top_of_set S) (S \<inter> f -` U) \<longleftrightarrow>
   684            openin (top_of_set (f ` S)) U"
   685 apply (rule continuous_right_inverse_imp_quotient_map)
   686 using assms apply force+
   687 done
   688 
   689 lemma continuous_imp_quotient_map:
   690   fixes f :: "'a::t2_space \<Rightarrow> 'b::t2_space"
   691   assumes "continuous_on S f" "f ` S = T" "compact S" "U \<subseteq> T"
   692     shows "openin (top_of_set S) (S \<inter> f -` U) \<longleftrightarrow>
   693            openin (top_of_set T) U"
   694   by (metis (no_types, lifting) assms closed_map_imp_quotient_map continuous_imp_closed_map)
   695 
   696 subsection%unimportant\<open>Pasting lemmas for functions, for of casewise definitions\<close>
   697 
   698 subsubsection\<open>on open sets\<close>
   699 
   700 lemma pasting_lemma:
   701   assumes ope: "\<And>i. i \<in> I \<Longrightarrow> openin X (T i)"
   702       and cont: "\<And>i. i \<in> I \<Longrightarrow> continuous_map(subtopology X (T i)) Y (f i)"
   703       and f: "\<And>i j x. \<lbrakk>i \<in> I; j \<in> I; x \<in> topspace X \<inter> T i \<inter> T j\<rbrakk> \<Longrightarrow> f i x = f j x"
   704       and g: "\<And>x. x \<in> topspace X \<Longrightarrow> \<exists>j. j \<in> I \<and> x \<in> T j \<and> g x = f j x"
   705     shows "continuous_map X Y g"
   706   unfolding continuous_map_openin_preimage_eq
   707 proof (intro conjI allI impI)
   708   show "g ` topspace X \<subseteq> topspace Y"
   709     using g cont continuous_map_image_subset_topspace topspace_subtopology by fastforce
   710 next
   711   fix U
   712   assume Y: "openin Y U"
   713   have T: "T i \<subseteq> topspace X" if "i \<in> I" for i
   714     using ope by (simp add: openin_subset that)
   715   have *: "topspace X \<inter> g -` U = (\<Union>i \<in> I. T i \<inter> f i -` U)"
   716     using f g T by fastforce
   717   have "\<And>i. i \<in> I \<Longrightarrow> openin X (T i \<inter> f i -` U)"
   718     using cont unfolding continuous_map_openin_preimage_eq
   719     by (metis Y T inf.commute inf_absorb1 ope topspace_subtopology openin_trans_full)
   720   then show "openin X (topspace X \<inter> g -` U)"
   721     by (auto simp: *)
   722 qed
   723 
   724 lemma pasting_lemma_exists:
   725   assumes X: "topspace X \<subseteq> (\<Union>i \<in> I. T i)"
   726       and ope: "\<And>i. i \<in> I \<Longrightarrow> openin X (T i)"
   727       and cont: "\<And>i. i \<in> I \<Longrightarrow> continuous_map (subtopology X (T i)) Y (f i)"
   728       and f: "\<And>i j x. \<lbrakk>i \<in> I; j \<in> I; x \<in> topspace X \<inter> T i \<inter> T j\<rbrakk> \<Longrightarrow> f i x = f j x"
   729     obtains g where "continuous_map X Y g" "\<And>x i. \<lbrakk>i \<in> I; x \<in> topspace X \<inter> T i\<rbrakk> \<Longrightarrow> g x = f i x"
   730 proof
   731   let ?h = "\<lambda>x. f (SOME i. i \<in> I \<and> x \<in> T i) x"
   732   show "continuous_map X Y ?h"
   733     apply (rule pasting_lemma [OF ope cont])
   734      apply (blast intro: f)+
   735     by (metis (no_types, lifting) UN_E X subsetD someI_ex)
   736   show "f (SOME i. i \<in> I \<and> x \<in> T i) x = f i x" if "i \<in> I" "x \<in> topspace X \<inter> T i" for i x
   737     by (metis (no_types, lifting) IntD2 IntI f someI_ex that)
   738 qed
   739 
   740 lemma pasting_lemma_locally_finite:
   741   assumes fin: "\<And>x. x \<in> topspace X \<Longrightarrow> \<exists>V. openin X V \<and> x \<in> V \<and> finite {i \<in> I. T i \<inter> V \<noteq> {}}"
   742     and clo: "\<And>i. i \<in> I \<Longrightarrow> closedin X (T i)"
   743     and cont:  "\<And>i. i \<in> I \<Longrightarrow> continuous_map(subtopology X (T i)) Y (f i)"
   744     and f: "\<And>i j x. \<lbrakk>i \<in> I; j \<in> I; x \<in> topspace X \<inter> T i \<inter> T j\<rbrakk> \<Longrightarrow> f i x = f j x"
   745     and g: "\<And>x. x \<in> topspace X \<Longrightarrow> \<exists>j. j \<in> I \<and> x \<in> T j \<and> g x = f j x"
   746   shows "continuous_map X Y g"
   747   unfolding continuous_map_closedin_preimage_eq
   748 proof (intro conjI allI impI)
   749   show "g ` topspace X \<subseteq> topspace Y"
   750     using g cont continuous_map_image_subset_topspace topspace_subtopology by fastforce
   751 next
   752   fix U
   753   assume Y: "closedin Y U"
   754   have T: "T i \<subseteq> topspace X" if "i \<in> I" for i
   755     using clo by (simp add: closedin_subset that)
   756   have *: "topspace X \<inter> g -` U = (\<Union>i \<in> I. T i \<inter> f i -` U)"
   757     using f g T by fastforce
   758   have cTf: "\<And>i. i \<in> I \<Longrightarrow> closedin X (T i \<inter> f i -` U)"
   759     using cont unfolding continuous_map_closedin_preimage_eq topspace_subtopology
   760     by (simp add: Int_absorb1 T Y clo closedin_closed_subtopology)
   761   have sub: "{Z \<in> (\<lambda>i. T i \<inter> f i -` U) ` I. Z \<inter> V \<noteq> {}}
   762            \<subseteq> (\<lambda>i. T i \<inter> f i -` U) ` {i \<in> I. T i \<inter> V \<noteq> {}}" for V
   763     by auto
   764   have 1: "(\<Union>i\<in>I. T i \<inter> f i -` U) \<subseteq> topspace X"
   765     using T by blast
   766   then have lf: "locally_finite_in X ((\<lambda>i. T i \<inter> f i -` U) ` I)"
   767     unfolding locally_finite_in_def
   768     using finite_subset [OF sub] fin by force
   769   show "closedin X (topspace X \<inter> g -` U)"
   770     apply (subst *)
   771     apply (rule closedin_locally_finite_Union)
   772      apply (auto intro: cTf lf)
   773     done
   774 qed
   775 
   776 subsubsection\<open>Likewise on closed sets, with a finiteness assumption\<close>
   777 
   778 lemma pasting_lemma_closed:
   779   assumes fin: "finite I"
   780     and clo: "\<And>i. i \<in> I \<Longrightarrow> closedin X (T i)"
   781     and cont:  "\<And>i. i \<in> I \<Longrightarrow> continuous_map(subtopology X (T i)) Y (f i)"
   782     and f: "\<And>i j x. \<lbrakk>i \<in> I; j \<in> I; x \<in> topspace X \<inter> T i \<inter> T j\<rbrakk> \<Longrightarrow> f i x = f j x"
   783     and g: "\<And>x. x \<in> topspace X \<Longrightarrow> \<exists>j. j \<in> I \<and> x \<in> T j \<and> g x = f j x"
   784   shows "continuous_map X Y g"
   785   using pasting_lemma_locally_finite [OF _ clo cont f g] fin by auto
   786 
   787 lemma pasting_lemma_exists_locally_finite:
   788   assumes fin: "\<And>x. x \<in> topspace X \<Longrightarrow> \<exists>V. openin X V \<and> x \<in> V \<and> finite {i \<in> I. T i \<inter> V \<noteq> {}}"
   789     and X: "topspace X \<subseteq> \<Union>(T ` I)"
   790     and clo: "\<And>i. i \<in> I \<Longrightarrow> closedin X (T i)"
   791     and cont:  "\<And>i. i \<in> I \<Longrightarrow> continuous_map(subtopology X (T i)) Y (f i)"
   792     and f: "\<And>i j x. \<lbrakk>i \<in> I; j \<in> I; x \<in> topspace X \<inter> T i \<inter> T j\<rbrakk> \<Longrightarrow> f i x = f j x"
   793     and g: "\<And>x. x \<in> topspace X \<Longrightarrow> \<exists>j. j \<in> I \<and> x \<in> T j \<and> g x = f j x"
   794   obtains g where "continuous_map X Y g" "\<And>x i. \<lbrakk>i \<in> I; x \<in> topspace X \<inter> T i\<rbrakk> \<Longrightarrow> g x = f i x"
   795 proof
   796   show "continuous_map X Y (\<lambda>x. f(@i. i \<in> I \<and> x \<in> T i) x)"
   797     apply (rule pasting_lemma_locally_finite [OF fin])
   798         apply (blast intro: assms)+
   799     by (metis (no_types, lifting) UN_E X set_rev_mp someI_ex)
   800 next
   801   fix x i
   802   assume "i \<in> I" and "x \<in> topspace X \<inter> T i"
   803   show "f (SOME i. i \<in> I \<and> x \<in> T i) x = f i x"
   804     apply (rule someI2_ex)
   805     using \<open>i \<in> I\<close> \<open>x \<in> topspace X \<inter> T i\<close> apply blast
   806     by (meson Int_iff \<open>i \<in> I\<close> \<open>x \<in> topspace X \<inter> T i\<close> f)
   807 qed
   808 
   809 lemma pasting_lemma_exists_closed:
   810   assumes fin: "finite I"
   811     and X: "topspace X \<subseteq> \<Union>(T ` I)"
   812     and clo: "\<And>i. i \<in> I \<Longrightarrow> closedin X (T i)"
   813     and cont:  "\<And>i. i \<in> I \<Longrightarrow> continuous_map(subtopology X (T i)) Y (f i)"
   814     and f: "\<And>i j x. \<lbrakk>i \<in> I; j \<in> I; x \<in> topspace X \<inter> T i \<inter> T j\<rbrakk> \<Longrightarrow> f i x = f j x"
   815   obtains g where "continuous_map X Y g" "\<And>x i. \<lbrakk>i \<in> I; x \<in> topspace X \<inter> T i\<rbrakk> \<Longrightarrow> g x = f i x"
   816 proof
   817   show "continuous_map X Y (\<lambda>x. f (SOME i. i \<in> I \<and> x \<in> T i) x)"
   818     apply (rule pasting_lemma_closed [OF \<open>finite I\<close> clo cont])
   819      apply (blast intro: f)+
   820     by (metis (mono_tags, lifting) UN_iff X someI_ex subset_iff)
   821 next
   822   fix x i
   823   assume "i \<in> I" "x \<in> topspace X \<inter> T i"
   824   then show "f (SOME i. i \<in> I \<and> x \<in> T i) x = f i x"
   825     by (metis (no_types, lifting) IntD2 IntI f someI_ex)
   826 qed
   827 
   828 lemma continuous_map_cases:
   829   assumes f: "continuous_map (subtopology X (X closure_of {x. P x})) Y f"
   830       and g: "continuous_map (subtopology X (X closure_of {x. \<not> P x})) Y g"
   831       and fg: "\<And>x. x \<in> X frontier_of {x. P x} \<Longrightarrow> f x = g x"
   832   shows "continuous_map X Y (\<lambda>x. if P x then f x else g x)"
   833 proof (rule pasting_lemma_closed)
   834   let ?f = "\<lambda>b. if b then f else g"
   835   let ?g = "\<lambda>x. if P x then f x else g x"
   836   let ?T = "\<lambda>b. if b then X closure_of {x. P x} else X closure_of {x. ~P x}"
   837   show "finite {True,False}" by auto
   838   have eq: "topspace X - Collect P = topspace X \<inter> {x. \<not> P x}"
   839     by blast
   840   show "?f i x = ?f j x"
   841     if "i \<in> {True,False}" "j \<in> {True,False}" and x: "x \<in> topspace X \<inter> ?T i \<inter> ?T j" for i j x
   842   proof -
   843     have "f x = g x"
   844       if "i" "\<not> j"
   845       apply (rule fg)
   846       unfolding frontier_of_closures eq
   847       using x that closure_of_restrict by fastforce
   848     moreover
   849     have "g x = f x"
   850       if "x \<in> X closure_of {x. \<not> P x}" "x \<in> X closure_of Collect P" "\<not> i" "j" for x
   851         apply (rule fg [symmetric])
   852         unfolding frontier_of_closures eq
   853         using x that closure_of_restrict by fastforce
   854     ultimately show ?thesis
   855       using that by (auto simp flip: closure_of_restrict)
   856   qed
   857   show "\<exists>j. j \<in> {True,False} \<and> x \<in> ?T j \<and> (if P x then f x else g x) = ?f j x"
   858     if "x \<in> topspace X" for x
   859     apply simp
   860     apply safe
   861     apply (metis Int_iff closure_of inf_sup_absorb mem_Collect_eq that)
   862     by (metis DiffI eq closure_of_subset_Int contra_subsetD mem_Collect_eq that)
   863 qed (auto simp: f g)
   864 
   865 lemma continuous_map_cases_alt:
   866   assumes f: "continuous_map (subtopology X (X closure_of {x \<in> topspace X. P x})) Y f"
   867       and g: "continuous_map (subtopology X (X closure_of {x \<in> topspace X. ~P x})) Y g"
   868       and fg: "\<And>x. x \<in> X frontier_of {x \<in> topspace X. P x} \<Longrightarrow> f x = g x"
   869     shows "continuous_map X Y (\<lambda>x. if P x then f x else g x)"
   870   apply (rule continuous_map_cases)
   871   using assms
   872     apply (simp_all add: Collect_conj_eq closure_of_restrict [symmetric] frontier_of_restrict [symmetric])
   873   done
   874 
   875 lemma continuous_map_cases_function:
   876   assumes contp: "continuous_map X Z p"
   877     and contf: "continuous_map (subtopology X {x \<in> topspace X. p x \<in> Z closure_of U}) Y f"
   878     and contg: "continuous_map (subtopology X {x \<in> topspace X. p x \<in> Z closure_of (topspace Z - U)}) Y g"
   879     and fg: "\<And>x. \<lbrakk>x \<in> topspace X; p x \<in> Z frontier_of U\<rbrakk> \<Longrightarrow> f x = g x"
   880   shows "continuous_map X Y (\<lambda>x. if p x \<in> U then f x else g x)"
   881 proof (rule continuous_map_cases_alt)
   882   show "continuous_map (subtopology X (X closure_of {x \<in> topspace X. p x \<in> U})) Y f"
   883   proof (rule continuous_map_from_subtopology_mono)
   884     let ?T = "{x \<in> topspace X. p x \<in> Z closure_of U}"
   885     show "continuous_map (subtopology X ?T) Y f"
   886       by (simp add: contf)
   887     show "X closure_of {x \<in> topspace X. p x \<in> U} \<subseteq> ?T"
   888       by (rule continuous_map_closure_preimage_subset [OF contp])
   889   qed
   890   show "continuous_map (subtopology X (X closure_of {x \<in> topspace X. p x \<notin> U})) Y g"
   891   proof (rule continuous_map_from_subtopology_mono)
   892     let ?T = "{x \<in> topspace X. p x \<in> Z closure_of (topspace Z - U)}"
   893     show "continuous_map (subtopology X ?T) Y g"
   894       by (simp add: contg)
   895     have "X closure_of {x \<in> topspace X. p x \<notin> U} \<subseteq> X closure_of {x \<in> topspace X. p x \<in> topspace Z - U}"
   896       apply (rule closure_of_mono)
   897       using continuous_map_closedin contp by fastforce
   898     then show "X closure_of {x \<in> topspace X. p x \<notin> U} \<subseteq> ?T"
   899       by (rule order_trans [OF _ continuous_map_closure_preimage_subset [OF contp]])
   900   qed
   901 next
   902   show "f x = g x" if "x \<in> X frontier_of {x \<in> topspace X. p x \<in> U}" for x
   903     using that continuous_map_frontier_frontier_preimage_subset [OF contp, of U] fg by blast
   904 qed
   905 
   906 subsection \<open>Retractions\<close>
   907 
   908 definition%important retraction :: "('a::topological_space) set \<Rightarrow> 'a set \<Rightarrow> ('a \<Rightarrow> 'a) \<Rightarrow> bool"
   909 where "retraction S T r \<longleftrightarrow>
   910   T \<subseteq> S \<and> continuous_on S r \<and> r ` S \<subseteq> T \<and> (\<forall>x\<in>T. r x = x)"
   911 
   912 definition%important retract_of (infixl "retract'_of" 50) where
   913 "T retract_of S  \<longleftrightarrow>  (\<exists>r. retraction S T r)"
   914 
   915 lemma retraction_idempotent: "retraction S T r \<Longrightarrow> x \<in> S \<Longrightarrow>  r (r x) = r x"
   916   unfolding retraction_def by auto
   917 
   918 text \<open>Preservation of fixpoints under (more general notion of) retraction\<close>
   919 
   920 lemma invertible_fixpoint_property:
   921   fixes S :: "'a::topological_space set"
   922     and T :: "'b::topological_space set"
   923   assumes contt: "continuous_on T i"
   924     and "i ` T \<subseteq> S"
   925     and contr: "continuous_on S r"
   926     and "r ` S \<subseteq> T"
   927     and ri: "\<And>y. y \<in> T \<Longrightarrow> r (i y) = y"
   928     and FP: "\<And>f. \<lbrakk>continuous_on S f; f ` S \<subseteq> S\<rbrakk> \<Longrightarrow> \<exists>x\<in>S. f x = x"
   929     and contg: "continuous_on T g"
   930     and "g ` T \<subseteq> T"
   931   obtains y where "y \<in> T" and "g y = y"
   932 proof -
   933   have "\<exists>x\<in>S. (i \<circ> g \<circ> r) x = x"
   934   proof (rule FP)
   935     show "continuous_on S (i \<circ> g \<circ> r)"
   936       by (meson contt contr assms(4) contg assms(8) continuous_on_compose continuous_on_subset)
   937     show "(i \<circ> g \<circ> r) ` S \<subseteq> S"
   938       using assms(2,4,8) by force
   939   qed
   940   then obtain x where x: "x \<in> S" "(i \<circ> g \<circ> r) x = x" ..
   941   then have *: "g (r x) \<in> T"
   942     using assms(4,8) by auto
   943   have "r ((i \<circ> g \<circ> r) x) = r x"
   944     using x by auto
   945   then show ?thesis
   946     using "*" ri that by auto
   947 qed
   948 
   949 lemma homeomorphic_fixpoint_property:
   950   fixes S :: "'a::topological_space set"
   951     and T :: "'b::topological_space set"
   952   assumes "S homeomorphic T"
   953   shows "(\<forall>f. continuous_on S f \<and> f ` S \<subseteq> S \<longrightarrow> (\<exists>x\<in>S. f x = x)) \<longleftrightarrow>
   954          (\<forall>g. continuous_on T g \<and> g ` T \<subseteq> T \<longrightarrow> (\<exists>y\<in>T. g y = y))"
   955          (is "?lhs = ?rhs")
   956 proof -
   957   obtain r i where r:
   958       "\<forall>x\<in>S. i (r x) = x" "r ` S = T" "continuous_on S r"
   959       "\<forall>y\<in>T. r (i y) = y" "i ` T = S" "continuous_on T i"
   960     using assms unfolding homeomorphic_def homeomorphism_def  by blast
   961   show ?thesis
   962   proof
   963     assume ?lhs
   964     with r show ?rhs
   965       by (metis invertible_fixpoint_property[of T i S r] order_refl)
   966   next
   967     assume ?rhs
   968     with r show ?lhs
   969       by (metis invertible_fixpoint_property[of S r T i] order_refl)
   970   qed
   971 qed
   972 
   973 lemma retract_fixpoint_property:
   974   fixes f :: "'a::topological_space \<Rightarrow> 'b::topological_space"
   975     and S :: "'a set"
   976   assumes "T retract_of S"
   977     and FP: "\<And>f. \<lbrakk>continuous_on S f; f ` S \<subseteq> S\<rbrakk> \<Longrightarrow> \<exists>x\<in>S. f x = x"
   978     and contg: "continuous_on T g"
   979     and "g ` T \<subseteq> T"
   980   obtains y where "y \<in> T" and "g y = y"
   981 proof -
   982   obtain h where "retraction S T h"
   983     using assms(1) unfolding retract_of_def ..
   984   then show ?thesis
   985     unfolding retraction_def
   986     using invertible_fixpoint_property[OF continuous_on_id _ _ _ _ FP]
   987     by (metis assms(4) contg image_ident that)
   988 qed
   989 
   990 lemma retraction:
   991   "retraction S T r \<longleftrightarrow>
   992     T \<subseteq> S \<and> continuous_on S r \<and> r ` S = T \<and> (\<forall>x \<in> T. r x = x)"
   993   by (force simp: retraction_def)
   994 
   995 lemma retractionE: \<comment> \<open>yields properties normalized wrt. simp -- less likely to loop\<close>
   996   assumes "retraction S T r"
   997   obtains "T = r ` S" "r ` S \<subseteq> S" "continuous_on S r" "\<And>x. x \<in> S \<Longrightarrow> r (r x) = r x"
   998 proof (rule that)
   999   from retraction [of S T r] assms
  1000   have "T \<subseteq> S" "continuous_on S r" "r ` S = T" and "\<forall>x \<in> T. r x = x"
  1001     by simp_all
  1002   then show "T = r ` S" "r ` S \<subseteq> S" "continuous_on S r"
  1003     by simp_all
  1004   from \<open>\<forall>x \<in> T. r x = x\<close> have "r x = x" if "x \<in> T" for x
  1005     using that by simp
  1006   with \<open>r ` S = T\<close> show "r (r x) = r x" if "x \<in> S" for x
  1007     using that by auto
  1008 qed
  1009 
  1010 lemma retract_ofE: \<comment> \<open>yields properties normalized wrt. simp -- less likely to loop\<close>
  1011   assumes "T retract_of S"
  1012   obtains r where "T = r ` S" "r ` S \<subseteq> S" "continuous_on S r" "\<And>x. x \<in> S \<Longrightarrow> r (r x) = r x"
  1013 proof -
  1014   from assms obtain r where "retraction S T r"
  1015     by (auto simp add: retract_of_def)
  1016   with that show thesis
  1017     by (auto elim: retractionE)
  1018 qed
  1019 
  1020 lemma retract_of_imp_extensible:
  1021   assumes "S retract_of T" and "continuous_on S f" and "f ` S \<subseteq> U"
  1022   obtains g where "continuous_on T g" "g ` T \<subseteq> U" "\<And>x. x \<in> S \<Longrightarrow> g x = f x"
  1023 proof -
  1024   from \<open>S retract_of T\<close> obtain r where "retraction T S r"
  1025     by (auto simp add: retract_of_def)
  1026   show thesis
  1027     by (rule that [of "f \<circ> r"])
  1028       (use \<open>continuous_on S f\<close> \<open>f ` S \<subseteq> U\<close> \<open>retraction T S r\<close> in \<open>auto simp: continuous_on_compose2 retraction\<close>)
  1029 qed
  1030 
  1031 lemma idempotent_imp_retraction:
  1032   assumes "continuous_on S f" and "f ` S \<subseteq> S" and "\<And>x. x \<in> S \<Longrightarrow> f(f x) = f x"
  1033     shows "retraction S (f ` S) f"
  1034 by (simp add: assms retraction)
  1035 
  1036 lemma retraction_subset:
  1037   assumes "retraction S T r" and "T \<subseteq> s'" and "s' \<subseteq> S"
  1038   shows "retraction s' T r"
  1039   unfolding retraction_def
  1040   by (metis assms continuous_on_subset image_mono retraction)
  1041 
  1042 lemma retract_of_subset:
  1043   assumes "T retract_of S" and "T \<subseteq> s'" and "s' \<subseteq> S"
  1044     shows "T retract_of s'"
  1045 by (meson assms retract_of_def retraction_subset)
  1046 
  1047 lemma retraction_refl [simp]: "retraction S S (\<lambda>x. x)"
  1048 by (simp add: retraction)
  1049 
  1050 lemma retract_of_refl [iff]: "S retract_of S"
  1051   unfolding retract_of_def retraction_def
  1052   using continuous_on_id by blast
  1053 
  1054 lemma retract_of_imp_subset:
  1055    "S retract_of T \<Longrightarrow> S \<subseteq> T"
  1056 by (simp add: retract_of_def retraction_def)
  1057 
  1058 lemma retract_of_empty [simp]:
  1059      "({} retract_of S) \<longleftrightarrow> S = {}"  "(S retract_of {}) \<longleftrightarrow> S = {}"
  1060 by (auto simp: retract_of_def retraction_def)
  1061 
  1062 lemma retract_of_singleton [iff]: "({x} retract_of S) \<longleftrightarrow> x \<in> S"
  1063   unfolding retract_of_def retraction_def by force
  1064 
  1065 lemma retraction_comp:
  1066    "\<lbrakk>retraction S T f; retraction T U g\<rbrakk>
  1067         \<Longrightarrow> retraction S U (g \<circ> f)"
  1068 apply (auto simp: retraction_def intro: continuous_on_compose2)
  1069 by blast
  1070 
  1071 lemma retract_of_trans [trans]:
  1072   assumes "S retract_of T" and "T retract_of U"
  1073     shows "S retract_of U"
  1074 using assms by (auto simp: retract_of_def intro: retraction_comp)
  1075 
  1076 lemma closedin_retract:
  1077   fixes S :: "'a :: t2_space set"
  1078   assumes "S retract_of T"
  1079     shows "closedin (top_of_set T) S"
  1080 proof -
  1081   obtain r where r: "S \<subseteq> T" "continuous_on T r" "r ` T \<subseteq> S" "\<And>x. x \<in> S \<Longrightarrow> r x = x"
  1082     using assms by (auto simp: retract_of_def retraction_def)
  1083   have "S = {x\<in>T. x = r x}"
  1084     using r by auto
  1085   also have "\<dots> = T \<inter> ((\<lambda>x. (x, r x)) -` ({y. \<exists>x. y = (x, x)}))"
  1086     unfolding vimage_def mem_Times_iff fst_conv snd_conv
  1087     using r
  1088     by auto
  1089   also have "closedin (top_of_set T) \<dots>"
  1090     by (rule continuous_closedin_preimage) (auto intro!: closed_diagonal continuous_on_Pair r)
  1091   finally show ?thesis .
  1092 qed
  1093 
  1094 lemma closedin_self [simp]: "closedin (top_of_set S) S"
  1095   by simp
  1096 
  1097 lemma retract_of_closed:
  1098     fixes S :: "'a :: t2_space set"
  1099     shows "\<lbrakk>closed T; S retract_of T\<rbrakk> \<Longrightarrow> closed S"
  1100   by (metis closedin_retract closedin_closed_eq)
  1101 
  1102 lemma retract_of_compact:
  1103      "\<lbrakk>compact T; S retract_of T\<rbrakk> \<Longrightarrow> compact S"
  1104   by (metis compact_continuous_image retract_of_def retraction)
  1105 
  1106 lemma retract_of_connected:
  1107     "\<lbrakk>connected T; S retract_of T\<rbrakk> \<Longrightarrow> connected S"
  1108   by (metis Topological_Spaces.connected_continuous_image retract_of_def retraction)
  1109 
  1110 lemma retraction_imp_quotient_map:
  1111   "openin (top_of_set S) (S \<inter> r -` U) \<longleftrightarrow> openin (top_of_set T) U"
  1112   if retraction: "retraction S T r" and "U \<subseteq> T"
  1113   using retraction apply (rule retractionE)
  1114   apply (rule continuous_right_inverse_imp_quotient_map [where g=r])
  1115   using \<open>U \<subseteq> T\<close> apply (auto elim: continuous_on_subset)
  1116   done
  1117 
  1118 lemma retract_of_Times:
  1119    "\<lbrakk>S retract_of s'; T retract_of t'\<rbrakk> \<Longrightarrow> (S \<times> T) retract_of (s' \<times> t')"
  1120 apply (simp add: retract_of_def retraction_def Sigma_mono, clarify)
  1121 apply (rename_tac f g)
  1122 apply (rule_tac x="\<lambda>z. ((f \<circ> fst) z, (g \<circ> snd) z)" in exI)
  1123 apply (rule conjI continuous_intros | erule continuous_on_subset | force)+
  1124 done
  1125 
  1126 subsection\<open>Retractions on a topological space\<close>
  1127 
  1128 definition retract_of_space :: "'a set \<Rightarrow> 'a topology \<Rightarrow> bool" (infix "retract'_of'_space" 50)
  1129   where "S retract_of_space X
  1130          \<equiv> S \<subseteq> topspace X \<and> (\<exists>r. continuous_map X (subtopology X S) r \<and> (\<forall>x \<in> S. r x = x))"
  1131 
  1132 lemma retract_of_space_retraction_maps:
  1133    "S retract_of_space X \<longleftrightarrow> S \<subseteq> topspace X \<and> (\<exists>r. retraction_maps X (subtopology X S) r id)"
  1134   by (auto simp: retract_of_space_def retraction_maps_def)
  1135 
  1136 lemma retract_of_space_section_map:
  1137    "S retract_of_space X \<longleftrightarrow> S \<subseteq> topspace X \<and> section_map (subtopology X S) X id"
  1138   unfolding retract_of_space_def retraction_maps_def section_map_def
  1139   by (auto simp: continuous_map_from_subtopology)
  1140 
  1141 lemma retract_of_space_imp_subset:
  1142    "S retract_of_space X \<Longrightarrow> S \<subseteq> topspace X"
  1143   by (simp add: retract_of_space_def)
  1144 
  1145 lemma retract_of_space_topspace:
  1146    "topspace X retract_of_space X"
  1147   using retract_of_space_def by force
  1148 
  1149 lemma retract_of_space_empty [simp]:
  1150    "{} retract_of_space X \<longleftrightarrow> topspace X = {}"
  1151   by (auto simp: continuous_map_def retract_of_space_def)
  1152 
  1153 lemma retract_of_space_singleton [simp]:
  1154   "{a} retract_of_space X \<longleftrightarrow> a \<in> topspace X"
  1155 proof -
  1156   have "continuous_map X (subtopology X {a}) (\<lambda>x. a) \<and> (\<lambda>x. a) a = a" if "a \<in> topspace X"
  1157     using that by simp
  1158   then show ?thesis
  1159     by (force simp: retract_of_space_def)
  1160 qed
  1161 
  1162 lemma retract_of_space_clopen:
  1163   assumes "openin X S" "closedin X S" "S = {} \<Longrightarrow> topspace X = {}"
  1164   shows "S retract_of_space X"
  1165 proof (cases "S = {}")
  1166   case False
  1167   then obtain a where "a \<in> S"
  1168     by blast
  1169   show ?thesis
  1170     unfolding retract_of_space_def
  1171   proof (intro exI conjI)
  1172     show "S \<subseteq> topspace X"
  1173       by (simp add: assms closedin_subset)
  1174     have "continuous_map X X (\<lambda>x. if x \<in> S then x else a)"
  1175     proof (rule continuous_map_cases)
  1176       show "continuous_map (subtopology X (X closure_of {x. x \<in> S})) X (\<lambda>x. x)"
  1177         by (simp add: continuous_map_from_subtopology)
  1178       show "continuous_map (subtopology X (X closure_of {x. x \<notin> S})) X (\<lambda>x. a)"
  1179         using \<open>S \<subseteq> topspace X\<close> \<open>a \<in> S\<close> by force
  1180       show "x = a" if "x \<in> X frontier_of {x. x \<in> S}" for x
  1181         using assms that clopenin_eq_frontier_of by fastforce
  1182     qed
  1183     then show "continuous_map X (subtopology X S) (\<lambda>x. if x \<in> S then x else a)"
  1184       using \<open>S \<subseteq> topspace X\<close> \<open>a \<in> S\<close>  by (auto simp: continuous_map_in_subtopology)
  1185   qed auto
  1186 qed (use assms in auto)
  1187 
  1188 lemma retract_of_space_disjoint_union:
  1189   assumes "openin X S" "openin X T" and ST: "disjnt S T" "S \<union> T = topspace X" and "S = {} \<Longrightarrow> topspace X = {}"
  1190   shows "S retract_of_space X"
  1191 proof (rule retract_of_space_clopen)
  1192   have "S \<inter> T = {}"
  1193     by (meson ST disjnt_def)
  1194   then have "S = topspace X - T"
  1195     using ST by auto
  1196   then show "closedin X S"
  1197     using \<open>openin X T\<close> by blast
  1198 qed (auto simp: assms)
  1199 
  1200 lemma retraction_maps_section_image1:
  1201   assumes "retraction_maps X Y r s"
  1202   shows "s ` (topspace Y) retract_of_space X"
  1203   unfolding retract_of_space_section_map
  1204 proof
  1205   show "s ` topspace Y \<subseteq> topspace X"
  1206     using assms continuous_map_image_subset_topspace retraction_maps_def by blast
  1207   show "section_map (subtopology X (s ` topspace Y)) X id"
  1208     unfolding section_map_def
  1209     using assms retraction_maps_to_retract_maps by blast
  1210 qed
  1211 
  1212 lemma retraction_maps_section_image2:
  1213    "retraction_maps X Y r s
  1214         \<Longrightarrow> subtopology X (s ` (topspace Y)) homeomorphic_space Y"
  1215   using embedding_map_imp_homeomorphic_space homeomorphic_space_sym section_imp_embedding_map
  1216         section_map_def by blast
  1217 
  1218 subsection\<open>Paths and path-connectedness\<close>
  1219 
  1220 definition pathin :: "'a topology \<Rightarrow> (real \<Rightarrow> 'a) \<Rightarrow> bool" where
  1221    "pathin X g \<equiv> continuous_map (subtopology euclideanreal {0..1}) X g"
  1222 
  1223 lemma pathin_compose:
  1224      "\<lbrakk>pathin X g; continuous_map X Y f\<rbrakk> \<Longrightarrow> pathin Y (f \<circ> g)"
  1225    by (simp add: continuous_map_compose pathin_def)
  1226 
  1227 lemma pathin_subtopology:
  1228      "pathin (subtopology X S) g \<longleftrightarrow> pathin X g \<and> (\<forall>x \<in> {0..1}. g x \<in> S)"
  1229   by (auto simp: pathin_def continuous_map_in_subtopology)
  1230 
  1231 lemma pathin_const:
  1232    "pathin X (\<lambda>x. a) \<longleftrightarrow> a \<in> topspace X"
  1233   by (simp add: pathin_def)
  1234    
  1235 lemma path_start_in_topspace: "pathin X g \<Longrightarrow> g 0 \<in> topspace X"
  1236   by (force simp: pathin_def continuous_map)
  1237 
  1238 lemma path_finish_in_topspace: "pathin X g \<Longrightarrow> g 1 \<in> topspace X"
  1239   by (force simp: pathin_def continuous_map)
  1240 
  1241 lemma path_image_subset_topspace: "pathin X g \<Longrightarrow> g ` ({0..1}) \<subseteq> topspace X"
  1242   by (force simp: pathin_def continuous_map)
  1243 
  1244 definition path_connected_space :: "'a topology \<Rightarrow> bool"
  1245   where "path_connected_space X \<equiv> \<forall>x \<in> topspace X. \<forall> y \<in> topspace X. \<exists>g. pathin X g \<and> g 0 = x \<and> g 1 = y"
  1246 
  1247 definition path_connectedin :: "'a topology \<Rightarrow> 'a set \<Rightarrow> bool"
  1248   where "path_connectedin X S \<equiv> S \<subseteq> topspace X \<and> path_connected_space(subtopology X S)"
  1249 
  1250 lemma path_connectedin_absolute [simp]:
  1251      "path_connectedin (subtopology X S) S \<longleftrightarrow> path_connectedin X S"
  1252   by (simp add: path_connectedin_def subtopology_subtopology topspace_subtopology)
  1253 
  1254 lemma path_connectedin_subset_topspace:
  1255      "path_connectedin X S \<Longrightarrow> S \<subseteq> topspace X"
  1256   by (simp add: path_connectedin_def)
  1257 
  1258 lemma path_connectedin_subtopology:
  1259      "path_connectedin (subtopology X S) T \<longleftrightarrow> path_connectedin X T \<and> T \<subseteq> S"
  1260   by (auto simp: path_connectedin_def subtopology_subtopology topspace_subtopology inf.absorb2)
  1261 
  1262 lemma path_connectedin:
  1263      "path_connectedin X S \<longleftrightarrow>
  1264         S \<subseteq> topspace X \<and>
  1265         (\<forall>x \<in> S. \<forall>y \<in> S. \<exists>g. pathin X g \<and> g ` {0..1} \<subseteq> S \<and> g 0 = x \<and> g 1 = y)"
  1266   unfolding path_connectedin_def path_connected_space_def pathin_def continuous_map_in_subtopology
  1267   by (intro conj_cong refl ball_cong) (simp_all add: inf.absorb_iff2 topspace_subtopology)
  1268 
  1269 lemma path_connectedin_topspace:
  1270      "path_connectedin X (topspace X) \<longleftrightarrow> path_connected_space X"
  1271   by (simp add: path_connectedin_def)
  1272 
  1273 lemma path_connected_imp_connected_space:
  1274   assumes "path_connected_space X"
  1275   shows "connected_space X"
  1276 proof -
  1277   have *: "\<exists>S. connectedin X S \<and> g 0 \<in> S \<and> g 1 \<in> S" if "pathin X g" for g
  1278   proof (intro exI conjI)
  1279     have "continuous_map (subtopology euclideanreal {0..1}) X g"
  1280       using connectedin_absolute that by (simp add: pathin_def)
  1281     then show "connectedin X (g ` {0..1})"
  1282       by (rule connectedin_continuous_map_image) auto
  1283   qed auto
  1284   show ?thesis
  1285     using assms
  1286     by (auto intro: * simp add: path_connected_space_def connected_space_subconnected Ball_def)
  1287 qed
  1288 
  1289 lemma path_connectedin_imp_connectedin:
  1290      "path_connectedin X S \<Longrightarrow> connectedin X S"
  1291   by (simp add: connectedin_def path_connected_imp_connected_space path_connectedin_def)
  1292 
  1293 lemma path_connected_space_topspace_empty:
  1294      "topspace X = {} \<Longrightarrow> path_connected_space X"
  1295   by (simp add: path_connected_space_def)
  1296 
  1297 lemma path_connectedin_empty [simp]: "path_connectedin X {}"
  1298   by (simp add: path_connectedin)
  1299 
  1300 lemma path_connectedin_singleton [simp]: "path_connectedin X {a} \<longleftrightarrow> a \<in> topspace X"
  1301 proof
  1302   show "path_connectedin X {a} \<Longrightarrow> a \<in> topspace X"
  1303     by (simp add: path_connectedin)
  1304   show "a \<in> topspace X \<Longrightarrow> path_connectedin X {a}"
  1305     unfolding path_connectedin
  1306     using pathin_const by fastforce
  1307 qed
  1308 
  1309 lemma path_connectedin_continuous_map_image:
  1310   assumes f: "continuous_map X Y f" and S: "path_connectedin X S"
  1311   shows "path_connectedin Y (f ` S)"
  1312 proof -
  1313   have fX: "f ` (topspace X) \<subseteq> topspace Y"
  1314     by (metis f continuous_map_image_subset_topspace)
  1315   show ?thesis
  1316     unfolding path_connectedin
  1317   proof (intro conjI ballI; clarify?)
  1318     fix x
  1319     assume "x \<in> S"
  1320     show "f x \<in> topspace Y"
  1321       by (meson S fX \<open>x \<in> S\<close> image_subset_iff path_connectedin_subset_topspace set_mp)
  1322   next
  1323     fix x y
  1324     assume "x \<in> S" and "y \<in> S"
  1325     then obtain g where g: "pathin X g" "g ` {0..1} \<subseteq> S" "g 0 = x" "g 1 = y"
  1326       using S  by (force simp: path_connectedin)
  1327     show "\<exists>g. pathin Y g \<and> g ` {0..1} \<subseteq> f ` S \<and> g 0 = f x \<and> g 1 = f y"
  1328     proof (intro exI conjI)
  1329       show "pathin Y (f \<circ> g)"
  1330         using \<open>pathin X g\<close> f pathin_compose by auto
  1331     qed (use g in auto)
  1332   qed
  1333 qed
  1334 
  1335 lemma path_connectedin_discrete_topology:
  1336   "path_connectedin (discrete_topology U) S \<longleftrightarrow> S \<subseteq> U \<and> (\<exists>a. S \<subseteq> {a})"
  1337   apply safe
  1338   using path_connectedin_subset_topspace apply fastforce
  1339    apply (meson connectedin_discrete_topology path_connectedin_imp_connectedin)
  1340   using subset_singletonD by fastforce
  1341 
  1342 lemma path_connected_space_discrete_topology:
  1343    "path_connected_space (discrete_topology U) \<longleftrightarrow> (\<exists>a. U \<subseteq> {a})"
  1344   by (metis path_connectedin_discrete_topology path_connectedin_topspace path_connected_space_topspace_empty
  1345             subset_singletonD topspace_discrete_topology)
  1346 
  1347 
  1348 lemma homeomorphic_path_connected_space_imp:
  1349      "\<lbrakk>path_connected_space X; X homeomorphic_space Y\<rbrakk> \<Longrightarrow> path_connected_space Y"
  1350   unfolding homeomorphic_space_def homeomorphic_maps_def
  1351   by (metis (no_types, hide_lams) continuous_map_closedin continuous_map_image_subset_topspace imageI order_class.order.antisym path_connectedin_continuous_map_image path_connectedin_topspace subsetI)
  1352 
  1353 lemma homeomorphic_path_connected_space:
  1354    "X homeomorphic_space Y \<Longrightarrow> path_connected_space X \<longleftrightarrow> path_connected_space Y"
  1355   by (meson homeomorphic_path_connected_space_imp homeomorphic_space_sym)
  1356 
  1357 lemma homeomorphic_map_path_connectedness:
  1358   assumes "homeomorphic_map X Y f" "U \<subseteq> topspace X"
  1359   shows "path_connectedin Y (f ` U) \<longleftrightarrow> path_connectedin X U"
  1360   unfolding path_connectedin_def
  1361 proof (intro conj_cong homeomorphic_path_connected_space)
  1362   show "(f ` U \<subseteq> topspace Y) = (U \<subseteq> topspace X)"
  1363     using assms homeomorphic_imp_surjective_map by blast
  1364 next
  1365   assume "U \<subseteq> topspace X"
  1366   show "subtopology Y (f ` U) homeomorphic_space subtopology X U"
  1367     using assms unfolding homeomorphic_eq_everything_map
  1368     by (metis (no_types, hide_lams) assms homeomorphic_map_subtopologies homeomorphic_space homeomorphic_space_sym image_mono inf.absorb_iff2)
  1369 qed
  1370 
  1371 lemma homeomorphic_map_path_connectedness_eq:
  1372    "homeomorphic_map X Y f \<Longrightarrow> path_connectedin X U \<longleftrightarrow> U \<subseteq> topspace X \<and> path_connectedin Y (f ` U)"
  1373   by (meson homeomorphic_map_path_connectedness path_connectedin_def)
  1374 
  1375 subsection\<open>Connected components\<close>
  1376 
  1377 definition connected_component_of :: "'a topology \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"
  1378   where "connected_component_of X x y \<equiv>
  1379         \<exists>T. connectedin X T \<and> x \<in> T \<and> y \<in> T"
  1380 
  1381 abbreviation connected_component_of_set
  1382   where "connected_component_of_set X x \<equiv> Collect (connected_component_of X x)"
  1383 
  1384 definition connected_components_of :: "'a topology \<Rightarrow> ('a set) set"
  1385   where "connected_components_of X \<equiv> connected_component_of_set X ` topspace X"
  1386 
  1387 lemma connected_component_in_topspace:
  1388    "connected_component_of X x y \<Longrightarrow> x \<in> topspace X \<and> y \<in> topspace X"
  1389   by (meson connected_component_of_def connectedin_subset_topspace in_mono)
  1390 
  1391 lemma connected_component_of_refl:
  1392    "connected_component_of X x x \<longleftrightarrow> x \<in> topspace X"
  1393   by (meson connected_component_in_topspace connected_component_of_def connectedin_sing insertI1)
  1394 
  1395 lemma connected_component_of_sym:
  1396    "connected_component_of X x y \<longleftrightarrow> connected_component_of X y x"
  1397   by (meson connected_component_of_def)
  1398 
  1399 lemma connected_component_of_trans:
  1400    "\<lbrakk>connected_component_of X x y; connected_component_of X y z\<rbrakk>
  1401         \<Longrightarrow> connected_component_of X x z"
  1402   unfolding connected_component_of_def
  1403   using connectedin_Un by fastforce
  1404 
  1405 lemma connected_component_of_mono:
  1406    "\<lbrakk>connected_component_of (subtopology X S) x y; S \<subseteq> T\<rbrakk>
  1407         \<Longrightarrow> connected_component_of (subtopology X T) x y"
  1408   by (metis connected_component_of_def connectedin_subtopology inf.absorb_iff2 subtopology_subtopology)
  1409 
  1410 lemma connected_component_of_set:
  1411    "connected_component_of_set X x = {y. \<exists>T. connectedin X T \<and> x \<in> T \<and> y \<in> T}"
  1412   by (meson connected_component_of_def)
  1413 
  1414 lemma connected_component_of_subset_topspace:
  1415    "connected_component_of_set X x \<subseteq> topspace X"
  1416   using connected_component_in_topspace by force
  1417 
  1418 lemma connected_component_of_eq_empty:
  1419    "connected_component_of_set X x = {} \<longleftrightarrow> (x \<notin> topspace X)"
  1420   using connected_component_in_topspace connected_component_of_refl by fastforce
  1421 
  1422 lemma connected_space_iff_connected_component:
  1423    "connected_space X \<longleftrightarrow> (\<forall>x \<in> topspace X. \<forall>y \<in> topspace X. connected_component_of X x y)"
  1424   by (simp add: connected_component_of_def connected_space_subconnected)
  1425 
  1426 lemma connected_space_imp_connected_component_of:
  1427    "\<lbrakk>connected_space X; a \<in> topspace X; b \<in> topspace X\<rbrakk>
  1428     \<Longrightarrow> connected_component_of X a b"
  1429   by (simp add: connected_space_iff_connected_component)
  1430 
  1431 lemma connected_space_connected_component_set:
  1432    "connected_space X \<longleftrightarrow> (\<forall>x \<in> topspace X. connected_component_of_set X x = topspace X)"
  1433   using connected_component_of_subset_topspace connected_space_iff_connected_component by fastforce
  1434 
  1435 lemma connected_component_of_maximal:
  1436    "\<lbrakk>connectedin X S; x \<in> S\<rbrakk> \<Longrightarrow> S \<subseteq> connected_component_of_set X x"
  1437   by (meson Ball_Collect connected_component_of_def)
  1438 
  1439 lemma connected_component_of_equiv:
  1440    "connected_component_of X x y \<longleftrightarrow>
  1441     x \<in> topspace X \<and> y \<in> topspace X \<and> connected_component_of X x = connected_component_of X y"
  1442   apply (simp add: connected_component_in_topspace fun_eq_iff)
  1443   by (meson connected_component_of_refl connected_component_of_sym connected_component_of_trans)
  1444 
  1445 lemma connected_component_of_disjoint:
  1446    "disjnt (connected_component_of_set X x) (connected_component_of_set X y)
  1447     \<longleftrightarrow> ~(connected_component_of X x y)"
  1448   using connected_component_of_equiv unfolding disjnt_iff by force
  1449 
  1450 lemma connected_component_of_eq:
  1451    "connected_component_of X x = connected_component_of X y \<longleftrightarrow>
  1452         (x \<notin> topspace X) \<and> (y \<notin> topspace X) \<or>
  1453         x \<in> topspace X \<and> y \<in> topspace X \<and>
  1454         connected_component_of X x y"
  1455   by (metis Collect_empty_eq_bot connected_component_of_eq_empty connected_component_of_equiv)
  1456 
  1457 lemma connectedin_connected_component_of:
  1458    "connectedin X (connected_component_of_set X x)"
  1459 proof -
  1460   have "connected_component_of_set X x = \<Union> {T. connectedin X T \<and> x \<in> T}"
  1461     by (auto simp: connected_component_of_def)
  1462   then show ?thesis
  1463     apply (rule ssubst)
  1464     by (blast intro: connectedin_Union)
  1465 qed
  1466 
  1467 
  1468 lemma Union_connected_components_of:
  1469    "\<Union>(connected_components_of X) = topspace X"
  1470   unfolding connected_components_of_def
  1471   apply (rule equalityI)
  1472   apply (simp add: SUP_least connected_component_of_subset_topspace)
  1473   using connected_component_of_refl by fastforce
  1474 
  1475 lemma connected_components_of_maximal:
  1476    "\<lbrakk>C \<in> connected_components_of X; connectedin X S; ~disjnt C S\<rbrakk> \<Longrightarrow> S \<subseteq> C"
  1477   unfolding connected_components_of_def disjnt_def
  1478   apply clarify
  1479   by (metis Int_emptyI connected_component_of_def connected_component_of_trans mem_Collect_eq)
  1480 
  1481 lemma pairwise_disjoint_connected_components_of:
  1482    "pairwise disjnt (connected_components_of X)"
  1483   unfolding connected_components_of_def pairwise_def
  1484   apply clarify
  1485   by (metis connected_component_of_disjoint connected_component_of_equiv)
  1486 
  1487 lemma complement_connected_components_of_Union:
  1488    "C \<in> connected_components_of X
  1489       \<Longrightarrow> topspace X - C = \<Union> (connected_components_of X - {C})"
  1490   apply (rule equalityI)
  1491   using Union_connected_components_of apply fastforce
  1492   by (metis Diff_cancel Diff_subset Union_connected_components_of cSup_singleton diff_Union_pairwise_disjoint equalityE insert_subsetI pairwise_disjoint_connected_components_of)
  1493 
  1494 lemma nonempty_connected_components_of:
  1495    "C \<in> connected_components_of X \<Longrightarrow> C \<noteq> {}"
  1496   unfolding connected_components_of_def
  1497   by (metis (no_types, lifting) connected_component_of_eq_empty imageE)
  1498 
  1499 lemma connected_components_of_subset:
  1500    "C \<in> connected_components_of X \<Longrightarrow> C \<subseteq> topspace X"
  1501   using Union_connected_components_of by fastforce
  1502 
  1503 lemma connectedin_connected_components_of:
  1504   assumes "C \<in> connected_components_of X"
  1505   shows "connectedin X C"
  1506 proof -
  1507   have "C \<in> connected_component_of_set X ` topspace X"
  1508     using assms connected_components_of_def by blast
  1509 then show ?thesis
  1510   using connectedin_connected_component_of by fastforce
  1511 qed
  1512 
  1513 lemma connected_component_in_connected_components_of:
  1514    "connected_component_of_set X a \<in> connected_components_of X \<longleftrightarrow> a \<in> topspace X"
  1515   apply (rule iffI)
  1516   using connected_component_of_eq_empty nonempty_connected_components_of apply fastforce
  1517   by (simp add: connected_components_of_def)
  1518 
  1519 lemma connected_space_iff_components_eq:
  1520    "connected_space X \<longleftrightarrow> (\<forall>C \<in> connected_components_of X. \<forall>C' \<in> connected_components_of X. C = C')"
  1521   apply (rule iffI)
  1522   apply (force simp: connected_components_of_def connected_space_connected_component_set image_iff)
  1523   by (metis connected_component_in_connected_components_of connected_component_of_refl connected_space_iff_connected_component mem_Collect_eq)
  1524 
  1525 lemma connected_components_of_eq_empty:
  1526    "connected_components_of X = {} \<longleftrightarrow> topspace X = {}"
  1527   by (simp add: connected_components_of_def)
  1528 
  1529 lemma connected_components_of_empty_space:
  1530    "topspace X = {} \<Longrightarrow> connected_components_of X = {}"
  1531   by (simp add: connected_components_of_eq_empty)
  1532 
  1533 lemma connected_components_of_subset_sing:
  1534    "connected_components_of X \<subseteq> {S} \<longleftrightarrow> connected_space X \<and> (topspace X = {} \<or> topspace X = S)"
  1535 proof (cases "topspace X = {}")
  1536   case True
  1537   then show ?thesis
  1538     by (simp add: connected_components_of_empty_space connected_space_topspace_empty)
  1539 next
  1540   case False
  1541   then show ?thesis
  1542     by (metis (no_types, hide_lams) Union_connected_components_of ccpo_Sup_singleton
  1543         connected_components_of_eq_empty connected_space_iff_components_eq insertI1 singletonD
  1544         subsetI subset_singleton_iff)
  1545 qed
  1546 
  1547 lemma connected_space_iff_components_subset_singleton:
  1548    "connected_space X \<longleftrightarrow> (\<exists>a. connected_components_of X \<subseteq> {a})"
  1549   by (simp add: connected_components_of_subset_sing)
  1550 
  1551 lemma connected_components_of_eq_singleton:
  1552    "connected_components_of X = {S}
  1553 \<longleftrightarrow> connected_space X \<and> topspace X \<noteq> {} \<and> S = topspace X"
  1554   by (metis ccpo_Sup_singleton connected_components_of_subset_sing insert_not_empty subset_singleton_iff)
  1555 
  1556 lemma connected_components_of_connected_space:
  1557    "connected_space X \<Longrightarrow> connected_components_of X = (if topspace X = {} then {} else {topspace X})"
  1558   by (simp add: connected_components_of_eq_empty connected_components_of_eq_singleton)
  1559 
  1560 lemma exists_connected_component_of_superset:
  1561   assumes "connectedin X S" and ne: "topspace X \<noteq> {}"
  1562   shows "\<exists>C. C \<in> connected_components_of X \<and> S \<subseteq> C"
  1563 proof (cases "S = {}")
  1564   case True
  1565   then show ?thesis
  1566     using ne connected_components_of_def by blast
  1567 next
  1568   case False
  1569   then show ?thesis
  1570     by (meson all_not_in_conv assms(1) connected_component_in_connected_components_of connected_component_of_maximal connectedin_subset_topspace in_mono)
  1571 qed
  1572 
  1573 lemma closedin_connected_components_of:
  1574   assumes "C \<in> connected_components_of X"
  1575   shows   "closedin X C"
  1576 proof -
  1577   obtain x where "x \<in> topspace X" and x: "C = connected_component_of_set X x"
  1578     using assms by (auto simp: connected_components_of_def)
  1579   have "connected_component_of_set X x \<subseteq> topspace X"
  1580     by (simp add: connected_component_of_subset_topspace)
  1581   moreover have "X closure_of connected_component_of_set X x \<subseteq> connected_component_of_set X x"
  1582   proof (rule connected_component_of_maximal)
  1583     show "connectedin X (X closure_of connected_component_of_set X x)"
  1584       by (simp add: connectedin_closure_of connectedin_connected_component_of)
  1585     show "x \<in> X closure_of connected_component_of_set X x"
  1586       by (simp add: \<open>x \<in> topspace X\<close> closure_of connected_component_of_refl)
  1587   qed
  1588   ultimately
  1589   show ?thesis
  1590     using closure_of_subset_eq x by auto
  1591 qed
  1592 
  1593 lemma closedin_connected_component_of:
  1594    "closedin X (connected_component_of_set X x)"
  1595   by (metis closedin_connected_components_of closedin_empty connected_component_in_connected_components_of connected_component_of_eq_empty)
  1596 
  1597 lemma connected_component_of_eq_overlap:
  1598    "connected_component_of_set X x = connected_component_of_set X y \<longleftrightarrow>
  1599       (x \<notin> topspace X) \<and> (y \<notin> topspace X) \<or>
  1600       ~(connected_component_of_set X x \<inter> connected_component_of_set X y = {})"
  1601   using connected_component_of_equiv by fastforce
  1602 
  1603 lemma connected_component_of_nonoverlap:
  1604    "connected_component_of_set X x \<inter> connected_component_of_set X y = {} \<longleftrightarrow>
  1605      (x \<notin> topspace X) \<or> (y \<notin> topspace X) \<or>
  1606      ~(connected_component_of_set X x = connected_component_of_set X y)"
  1607   by (metis connected_component_of_eq_empty connected_component_of_eq_overlap inf.idem)
  1608 
  1609 lemma connected_component_of_overlap:
  1610    "~(connected_component_of_set X x \<inter> connected_component_of_set X y = {}) \<longleftrightarrow>
  1611     x \<in> topspace X \<and> y \<in> topspace X \<and>
  1612     connected_component_of_set X x = connected_component_of_set X y"
  1613   by (meson connected_component_of_nonoverlap)
  1614 
  1615 
  1616 end