src/HOL/Analysis/Product_Topology.thy
author wenzelm
Mon Mar 25 17:21:26 2019 +0100 (3 months ago)
changeset 69981 3dced198b9ec
parent 69945 35ba13ac6e5c
child 69986 f2d327275065
permissions -rw-r--r--
more strict AFP properties;
     1 section\<open>The binary product topology\<close>
     2 
     3 theory Product_Topology
     4 imports Function_Topology Abstract_Limits
     5 begin
     6 
     7 section \<open>Product Topology\<close> 
     8 
     9 subsection\<open>Definition\<close>
    10 
    11 definition prod_topology :: "'a topology \<Rightarrow> 'b topology \<Rightarrow> ('a \<times> 'b) topology" where
    12  "prod_topology X Y \<equiv> topology (arbitrary union_of (\<lambda>U. U \<in> {S \<times> T |S T. openin X S \<and> openin Y T}))"
    13 
    14 lemma open_product_open:
    15   assumes "open A"
    16   shows "\<exists>\<U>. \<U> \<subseteq> {S \<times> T |S T. open S \<and> open T} \<and> \<Union> \<U> = A"
    17 proof -
    18   obtain f g where *: "\<And>u. u \<in> A \<Longrightarrow> open (f u) \<and> open (g u) \<and> u \<in> (f u) \<times> (g u) \<and> (f u) \<times> (g u) \<subseteq> A"
    19     using open_prod_def [of A] assms by metis
    20   let ?\<U> = "(\<lambda>u. f u \<times> g u) ` A"
    21   show ?thesis
    22     by (rule_tac x="?\<U>" in exI) (auto simp: dest: *)
    23 qed
    24 
    25 lemma open_product_open_eq: "(arbitrary union_of (\<lambda>U. \<exists>S T. U = S \<times> T \<and> open S \<and> open T)) = open"
    26   by (force simp: union_of_def arbitrary_def intro: open_product_open open_Times)
    27 
    28 lemma openin_prod_topology:
    29    "openin (prod_topology X Y) = arbitrary union_of (\<lambda>U. U \<in> {S \<times> T |S T. openin X S \<and> openin Y T})"
    30   unfolding prod_topology_def
    31 proof (rule topology_inverse')
    32   show "istopology (arbitrary union_of (\<lambda>U. U \<in> {S \<times> T |S T. openin X S \<and> openin Y T}))"
    33     apply (rule istopology_base, simp)
    34     by (metis openin_Int Times_Int_Times)
    35 qed
    36 
    37 lemma topspace_prod_topology [simp]:
    38    "topspace (prod_topology X Y) = topspace X \<times> topspace Y"
    39 proof -
    40   have "topspace(prod_topology X Y) = \<Union> (Collect (openin (prod_topology X Y)))" (is "_ = ?Z")
    41     unfolding topspace_def ..
    42   also have "\<dots> = topspace X \<times> topspace Y"
    43   proof
    44     show "?Z \<subseteq> topspace X \<times> topspace Y"
    45       apply (auto simp: openin_prod_topology union_of_def arbitrary_def)
    46       using openin_subset by force+
    47   next
    48     have *: "\<exists>A B. topspace X \<times> topspace Y = A \<times> B \<and> openin X A \<and> openin Y B"
    49       by blast
    50     show "topspace X \<times> topspace Y \<subseteq> ?Z"
    51       apply (rule Union_upper)
    52       using * by (simp add: openin_prod_topology arbitrary_union_of_inc)
    53   qed
    54   finally show ?thesis .
    55 qed
    56 
    57 lemma subtopology_Times:
    58   shows "subtopology (prod_topology X Y) (S \<times> T) = prod_topology (subtopology X S) (subtopology Y T)"
    59 proof -
    60   have "((\<lambda>U. \<exists>S T. U = S \<times> T \<and> openin X S \<and> openin Y T) relative_to S \<times> T) =
    61         (\<lambda>U. \<exists>S' T'. U = S' \<times> T' \<and> (openin X relative_to S) S' \<and> (openin Y relative_to T) T')"
    62     by (auto simp: relative_to_def Times_Int_Times fun_eq_iff) metis
    63   then show ?thesis
    64     by (simp add: topology_eq openin_prod_topology arbitrary_union_of_relative_to flip: openin_relative_to)
    65 qed
    66 
    67 lemma prod_topology_subtopology:
    68     "prod_topology (subtopology X S) Y = subtopology (prod_topology X Y) (S \<times> topspace Y)"
    69     "prod_topology X (subtopology Y T) = subtopology (prod_topology X Y) (topspace X \<times> T)"
    70 by (auto simp: subtopology_Times)
    71 
    72 lemma prod_topology_discrete_topology:
    73      "discrete_topology (S \<times> T) = prod_topology (discrete_topology S) (discrete_topology T)"
    74   by (auto simp: discrete_topology_unique openin_prod_topology intro: arbitrary_union_of_inc)
    75 
    76 lemma prod_topology_euclidean [simp]: "prod_topology euclidean euclidean = euclidean"
    77   by (simp add: prod_topology_def open_product_open_eq)
    78 
    79 lemma prod_topology_subtopology_eu [simp]:
    80   "prod_topology (subtopology euclidean S) (subtopology euclidean T) = subtopology euclidean (S \<times> T)"
    81   by (simp add: prod_topology_subtopology subtopology_subtopology Times_Int_Times)
    82 
    83 lemma continuous_map_subtopology_eu [simp]:
    84   "continuous_map (subtopology euclidean S) (subtopology euclidean T) h \<longleftrightarrow> continuous_on S h \<and> h ` S \<subseteq> T"
    85   apply safe
    86   apply (metis continuous_map_closedin_preimage_eq continuous_map_in_subtopology continuous_on_closed order_refl topspace_euclidean_subtopology)
    87   apply (simp add: continuous_map_closedin_preimage_eq image_subset_iff)
    88   by (metis (no_types, hide_lams) continuous_map_closedin_preimage_eq continuous_map_in_subtopology continuous_on_closed order_refl topspace_euclidean_subtopology)
    89 
    90 lemma openin_prod_topology_alt:
    91      "openin (prod_topology X Y) S \<longleftrightarrow>
    92       (\<forall>x y. (x,y) \<in> S \<longrightarrow> (\<exists>U V. openin X U \<and> openin Y V \<and> x \<in> U \<and> y \<in> V \<and> U \<times> V \<subseteq> S))"
    93   apply (auto simp: openin_prod_topology arbitrary_union_of_alt, fastforce)
    94   by (metis mem_Sigma_iff)
    95 
    96 lemma open_map_fst: "open_map (prod_topology X Y) X fst"
    97   unfolding open_map_def openin_prod_topology_alt
    98   by (force simp: openin_subopen [of X "fst ` _"] intro: subset_fst_imageI)
    99 
   100 lemma open_map_snd: "open_map (prod_topology X Y) Y snd"
   101   unfolding open_map_def openin_prod_topology_alt
   102   by (force simp: openin_subopen [of Y "snd ` _"] intro: subset_snd_imageI)
   103 
   104 lemma openin_Times:
   105      "openin (prod_topology X Y) (S \<times> T) \<longleftrightarrow> S = {} \<or> T = {} \<or> openin X S \<and> openin Y T"
   106 proof (cases "S = {} \<or> T = {}")
   107   case False
   108   then show ?thesis
   109     apply (simp add: openin_prod_topology_alt openin_subopen [of X S] openin_subopen [of Y T] times_subset_iff, safe)
   110       apply (meson|force)+
   111     done
   112 qed force
   113 
   114 
   115 lemma closure_of_Times:
   116    "(prod_topology X Y) closure_of (S \<times> T) = (X closure_of S) \<times> (Y closure_of T)"  (is "?lhs = ?rhs")
   117 proof
   118   show "?lhs \<subseteq> ?rhs"
   119     by (clarsimp simp: closure_of_def openin_prod_topology_alt) blast
   120   show "?rhs \<subseteq> ?lhs"
   121     by (clarsimp simp: closure_of_def openin_prod_topology_alt) (meson SigmaI subsetD)
   122 qed
   123 
   124 lemma closedin_Times:
   125    "closedin (prod_topology X Y) (S \<times> T) \<longleftrightarrow> S = {} \<or> T = {} \<or> closedin X S \<and> closedin Y T"
   126   by (auto simp: closure_of_Times times_eq_iff simp flip: closure_of_eq)
   127 
   128 lemma interior_of_Times: "(prod_topology X Y) interior_of (S \<times> T) = (X interior_of S) \<times> (Y interior_of T)"
   129 proof (rule interior_of_unique)
   130   show "(X interior_of S) \<times> Y interior_of T \<subseteq> S \<times> T"
   131     by (simp add: Sigma_mono interior_of_subset)
   132   show "openin (prod_topology X Y) ((X interior_of S) \<times> Y interior_of T)"
   133     by (simp add: openin_Times)
   134 next
   135   show "T' \<subseteq> (X interior_of S) \<times> Y interior_of T" if "T' \<subseteq> S \<times> T" "openin (prod_topology X Y) T'" for T'
   136   proof (clarsimp; intro conjI)
   137     fix a :: "'a" and b :: "'b"
   138     assume "(a, b) \<in> T'"
   139     with that obtain U V where UV: "openin X U" "openin Y V" "a \<in> U" "b \<in> V" "U \<times> V \<subseteq> T'"
   140       by (metis openin_prod_topology_alt)
   141     then show "a \<in> X interior_of S"
   142       using interior_of_maximal_eq that(1) by fastforce
   143     show "b \<in> Y interior_of T"
   144       using UV interior_of_maximal_eq that(1)
   145       by (metis SigmaI mem_Sigma_iff subset_eq)
   146   qed
   147 qed
   148 
   149 subsection \<open>Continuity\<close>
   150 
   151 lemma continuous_map_pairwise:
   152    "continuous_map Z (prod_topology X Y) f \<longleftrightarrow> continuous_map Z X (fst \<circ> f) \<and> continuous_map Z Y (snd \<circ> f)"
   153    (is "?lhs = ?rhs")
   154 proof -
   155   let ?g = "fst \<circ> f" and ?h = "snd \<circ> f"
   156   have f: "f x = (?g x, ?h x)" for x
   157     by auto
   158   show ?thesis
   159   proof (cases "(\<forall>x \<in> topspace Z. ?g x \<in> topspace X) \<and> (\<forall>x \<in> topspace Z. ?h x \<in> topspace Y)")
   160     case True
   161     show ?thesis
   162     proof safe
   163       assume "continuous_map Z (prod_topology X Y) f"
   164       then have "openin Z {x \<in> topspace Z. fst (f x) \<in> U}" if "openin X U" for U
   165         unfolding continuous_map_def using True that
   166         apply clarify
   167         apply (drule_tac x="U \<times> topspace Y" in spec)
   168         by (simp add: openin_Times mem_Times_iff cong: conj_cong)
   169       with True show "continuous_map Z X (fst \<circ> f)"
   170         by (auto simp: continuous_map_def)
   171     next
   172       assume "continuous_map Z (prod_topology X Y) f"
   173       then have "openin Z {x \<in> topspace Z. snd (f x) \<in> V}" if "openin Y V" for V
   174         unfolding continuous_map_def using True that
   175         apply clarify
   176         apply (drule_tac x="topspace X \<times> V" in spec)
   177         by (simp add: openin_Times mem_Times_iff cong: conj_cong)
   178       with True show "continuous_map Z Y (snd \<circ> f)"
   179         by (auto simp: continuous_map_def)
   180     next
   181       assume Z: "continuous_map Z X (fst \<circ> f)" "continuous_map Z Y (snd \<circ> f)"
   182       have *: "openin Z {x \<in> topspace Z. f x \<in> W}"
   183         if "\<And>w. w \<in> W \<Longrightarrow> \<exists>U V. openin X U \<and> openin Y V \<and> w \<in> U \<times> V \<and> U \<times> V \<subseteq> W" for W
   184       proof (subst openin_subopen, clarify)
   185         fix x :: "'a"
   186         assume "x \<in> topspace Z" and "f x \<in> W"
   187         with that [OF \<open>f x \<in> W\<close>]
   188         obtain U V where UV: "openin X U" "openin Y V" "f x \<in> U \<times> V" "U \<times> V \<subseteq> W"
   189           by auto
   190         with Z  UV show "\<exists>T. openin Z T \<and> x \<in> T \<and> T \<subseteq> {x \<in> topspace Z. f x \<in> W}"
   191           apply (rule_tac x="{x \<in> topspace Z. ?g x \<in> U} \<inter> {x \<in> topspace Z. ?h x \<in> V}" in exI)
   192           apply (auto simp: \<open>x \<in> topspace Z\<close> continuous_map_def)
   193           done
   194       qed
   195       show "continuous_map Z (prod_topology X Y) f"
   196         using True by (simp add: continuous_map_def openin_prod_topology_alt mem_Times_iff *)
   197     qed
   198   qed (auto simp: continuous_map_def)
   199 qed
   200 
   201 lemma continuous_map_paired:
   202    "continuous_map Z (prod_topology X Y) (\<lambda>x. (f x,g x)) \<longleftrightarrow> continuous_map Z X f \<and> continuous_map Z Y g"
   203   by (simp add: continuous_map_pairwise o_def)
   204 
   205 lemma continuous_map_pairedI [continuous_intros]:
   206    "\<lbrakk>continuous_map Z X f; continuous_map Z Y g\<rbrakk> \<Longrightarrow> continuous_map Z (prod_topology X Y) (\<lambda>x. (f x,g x))"
   207   by (simp add: continuous_map_pairwise o_def)
   208 
   209 lemma continuous_map_fst [continuous_intros]: "continuous_map (prod_topology X Y) X fst"
   210   using continuous_map_pairwise [of "prod_topology X Y" X Y id]
   211   by (simp add: continuous_map_pairwise)
   212 
   213 lemma continuous_map_snd [continuous_intros]: "continuous_map (prod_topology X Y) Y snd"
   214   using continuous_map_pairwise [of "prod_topology X Y" X Y id]
   215   by (simp add: continuous_map_pairwise)
   216 
   217 lemma continuous_map_fst_of [continuous_intros]:
   218    "continuous_map Z (prod_topology X Y) f \<Longrightarrow> continuous_map Z X (fst \<circ> f)"
   219   by (simp add: continuous_map_pairwise)
   220 
   221 lemma continuous_map_snd_of [continuous_intros]:
   222    "continuous_map Z (prod_topology X Y) f \<Longrightarrow> continuous_map Z Y (snd \<circ> f)"
   223   by (simp add: continuous_map_pairwise)
   224 
   225 lemma continuous_map_if_iff [simp]: "continuous_map X Y (\<lambda>x. if P then f x else g x) \<longleftrightarrow> continuous_map X Y (if P then f else g)"
   226   by simp
   227 
   228 lemma continuous_map_if [continuous_intros]: "\<lbrakk>P \<Longrightarrow> continuous_map X Y f; ~P \<Longrightarrow> continuous_map X Y g\<rbrakk>
   229       \<Longrightarrow> continuous_map X Y (\<lambda>x. if P then f x else g x)"
   230   by simp
   231 
   232 lemma continuous_map_subtopology_fst [continuous_intros]: "continuous_map (subtopology (prod_topology X Y) Z) X fst"
   233       using continuous_map_from_subtopology continuous_map_fst by force
   234 
   235 lemma continuous_map_subtopology_snd [continuous_intros]: "continuous_map (subtopology (prod_topology X Y) Z) Y snd"
   236       using continuous_map_from_subtopology continuous_map_snd by force
   237 
   238 lemma quotient_map_fst [simp]:
   239    "quotient_map(prod_topology X Y) X fst \<longleftrightarrow> (topspace Y = {} \<longrightarrow> topspace X = {})"
   240   by (auto simp: continuous_open_quotient_map open_map_fst continuous_map_fst)
   241 
   242 lemma quotient_map_snd [simp]:
   243    "quotient_map(prod_topology X Y) Y snd \<longleftrightarrow> (topspace X = {} \<longrightarrow> topspace Y = {})"
   244   by (auto simp: continuous_open_quotient_map open_map_snd continuous_map_snd)
   245 
   246 lemma retraction_map_fst:
   247    "retraction_map (prod_topology X Y) X fst \<longleftrightarrow> (topspace Y = {} \<longrightarrow> topspace X = {})"
   248 proof (cases "topspace Y = {}")
   249   case True
   250   then show ?thesis
   251     using continuous_map_image_subset_topspace
   252     by (fastforce simp: retraction_map_def retraction_maps_def continuous_map_fst continuous_map_on_empty)
   253 next
   254   case False
   255   have "\<exists>g. continuous_map X (prod_topology X Y) g \<and> (\<forall>x\<in>topspace X. fst (g x) = x)"
   256     if y: "y \<in> topspace Y" for y
   257     by (rule_tac x="\<lambda>x. (x,y)" in exI) (auto simp: y continuous_map_paired)
   258   with False have "retraction_map (prod_topology X Y) X fst"
   259     by (fastforce simp: retraction_map_def retraction_maps_def continuous_map_fst)
   260   with False show ?thesis
   261     by simp
   262 qed
   263 
   264 lemma retraction_map_snd:
   265    "retraction_map (prod_topology X Y) Y snd \<longleftrightarrow> (topspace X = {} \<longrightarrow> topspace Y = {})"
   266 proof (cases "topspace X = {}")
   267   case True
   268   then show ?thesis
   269     using continuous_map_image_subset_topspace
   270     by (fastforce simp: retraction_map_def retraction_maps_def continuous_map_fst continuous_map_on_empty)
   271 next
   272   case False
   273   have "\<exists>g. continuous_map Y (prod_topology X Y) g \<and> (\<forall>y\<in>topspace Y. snd (g y) = y)"
   274     if x: "x \<in> topspace X" for x
   275     by (rule_tac x="\<lambda>y. (x,y)" in exI) (auto simp: x continuous_map_paired)
   276   with False have "retraction_map (prod_topology X Y) Y snd"
   277     by (fastforce simp: retraction_map_def retraction_maps_def continuous_map_snd)
   278   with False show ?thesis
   279     by simp
   280 qed
   281 
   282 
   283 lemma continuous_map_of_fst:
   284    "continuous_map (prod_topology X Y) Z (f \<circ> fst) \<longleftrightarrow> topspace Y = {} \<or> continuous_map X Z f"
   285 proof (cases "topspace Y = {}")
   286   case True
   287   then show ?thesis
   288     by (simp add: continuous_map_on_empty)
   289 next
   290   case False
   291   then show ?thesis
   292     by (simp add: continuous_compose_quotient_map_eq quotient_map_fst)
   293 qed
   294 
   295 lemma continuous_map_of_snd:
   296    "continuous_map (prod_topology X Y) Z (f \<circ> snd) \<longleftrightarrow> topspace X = {} \<or> continuous_map Y Z f"
   297 proof (cases "topspace X = {}")
   298   case True
   299   then show ?thesis
   300     by (simp add: continuous_map_on_empty)
   301 next
   302   case False
   303   then show ?thesis
   304     by (simp add: continuous_compose_quotient_map_eq quotient_map_snd)
   305 qed
   306 
   307 lemma continuous_map_prod_top:
   308    "continuous_map (prod_topology X Y) (prod_topology X' Y') (\<lambda>(x,y). (f x, g y)) \<longleftrightarrow>
   309     topspace (prod_topology X Y) = {} \<or> continuous_map X X' f \<and> continuous_map Y Y' g"
   310 proof (cases "topspace (prod_topology X Y) = {}")
   311   case True
   312   then show ?thesis
   313     by (simp add: continuous_map_on_empty)
   314 next
   315   case False
   316   then show ?thesis
   317     by (simp add: continuous_map_paired case_prod_unfold continuous_map_of_fst [unfolded o_def] continuous_map_of_snd [unfolded o_def])
   318 qed
   319 
   320 lemma in_prod_topology_closure_of:
   321   assumes  "z \<in> (prod_topology X Y) closure_of S"
   322   shows "fst z \<in> X closure_of (fst ` S)" "snd z \<in> Y closure_of (snd ` S)"
   323   using assms continuous_map_eq_image_closure_subset continuous_map_fst apply fastforce
   324   using assms continuous_map_eq_image_closure_subset continuous_map_snd apply fastforce
   325   done
   326 
   327 
   328 proposition compact_space_prod_topology:
   329    "compact_space(prod_topology X Y) \<longleftrightarrow> topspace(prod_topology X Y) = {} \<or> compact_space X \<and> compact_space Y"
   330 proof (cases "topspace(prod_topology X Y) = {}")
   331   case True
   332   then show ?thesis
   333     using compact_space_topspace_empty by blast
   334 next
   335   case False
   336   then have non_mt: "topspace X \<noteq> {}" "topspace Y \<noteq> {}"
   337     by auto
   338   have "compact_space X" "compact_space Y" if "compact_space(prod_topology X Y)"
   339   proof -
   340     have "compactin X (fst ` (topspace X \<times> topspace Y))"
   341       by (metis compact_space_def continuous_map_fst image_compactin that topspace_prod_topology)
   342     moreover
   343     have "compactin Y (snd ` (topspace X \<times> topspace Y))"
   344       by (metis compact_space_def continuous_map_snd image_compactin that topspace_prod_topology)
   345     ultimately show "compact_space X" "compact_space Y"
   346       by (simp_all add: non_mt compact_space_def)
   347   qed
   348   moreover
   349   define \<X> where "\<X> \<equiv> (\<lambda>V. topspace X \<times> V) ` Collect (openin Y)"
   350   define \<Y> where "\<Y> \<equiv> (\<lambda>U. U \<times> topspace Y) ` Collect (openin X)"
   351   have "compact_space(prod_topology X Y)" if "compact_space X" "compact_space Y"
   352   proof (rule Alexander_subbase_alt)
   353     show toptop: "topspace X \<times> topspace Y \<subseteq> \<Union>(\<X> \<union> \<Y>)"
   354       unfolding \<X>_def \<Y>_def by auto
   355     fix \<C> :: "('a \<times> 'b) set set"
   356     assume \<C>: "\<C> \<subseteq> \<X> \<union> \<Y>" "topspace X \<times> topspace Y \<subseteq> \<Union>\<C>"
   357     then obtain \<X>' \<Y>' where XY: "\<X>' \<subseteq> \<X>" "\<Y>' \<subseteq> \<Y>" and \<C>eq: "\<C> = \<X>' \<union> \<Y>'"
   358       using subset_UnE by metis
   359     then have sub: "topspace X \<times> topspace Y \<subseteq> \<Union>(\<X>' \<union> \<Y>')"
   360       using \<C> by simp
   361     obtain \<U> \<V> where \<U>: "\<And>U. U \<in> \<U> \<Longrightarrow> openin X U" "\<Y>' = (\<lambda>U. U \<times> topspace Y) ` \<U>"
   362       and \<V>: "\<And>V. V \<in> \<V> \<Longrightarrow> openin Y V" "\<X>' = (\<lambda>V. topspace X \<times> V) ` \<V>"
   363       using XY by (clarsimp simp add: \<X>_def \<Y>_def subset_image_iff) (force simp add: subset_iff)
   364     have "\<exists>\<D>. finite \<D> \<and> \<D> \<subseteq> \<X>' \<union> \<Y>' \<and> topspace X \<times> topspace Y \<subseteq> \<Union> \<D>"
   365     proof -
   366       have "topspace X \<subseteq> \<Union>\<U> \<or> topspace Y \<subseteq> \<Union>\<V>"
   367         using \<U> \<V> \<C> \<C>eq by auto
   368       then have *: "\<exists>\<D>. finite \<D> \<and>
   369                (\<forall>x \<in> \<D>. x \<in> (\<lambda>V. topspace X \<times> V) ` \<V> \<or> x \<in> (\<lambda>U. U \<times> topspace Y) ` \<U>) \<and>
   370                (topspace X \<times> topspace Y \<subseteq> \<Union>\<D>)"
   371       proof
   372         assume "topspace X \<subseteq> \<Union>\<U>"
   373         with \<open>compact_space X\<close> \<U> obtain \<F> where "finite \<F>" "\<F> \<subseteq> \<U>" "topspace X \<subseteq> \<Union>\<F>"
   374           by (meson compact_space_alt)
   375         with that show ?thesis
   376           by (rule_tac x="(\<lambda>D. D \<times> topspace Y) ` \<F>" in exI) auto
   377       next
   378         assume "topspace Y \<subseteq> \<Union>\<V>"
   379         with \<open>compact_space Y\<close> \<V> obtain \<F> where "finite \<F>" "\<F> \<subseteq> \<V>" "topspace Y \<subseteq> \<Union>\<F>"
   380           by (meson compact_space_alt)
   381         with that show ?thesis
   382           by (rule_tac x="(\<lambda>C. topspace X \<times> C) ` \<F>" in exI) auto
   383       qed
   384       then show ?thesis
   385         using that \<U> \<V> by blast
   386     qed
   387     then show "\<exists>\<D>. finite \<D> \<and> \<D> \<subseteq> \<C> \<and> topspace X \<times> topspace Y \<subseteq> \<Union> \<D>"
   388       using \<C> \<C>eq by blast
   389   next
   390     have "(finite intersection_of (\<lambda>x. x \<in> \<X> \<or> x \<in> \<Y>) relative_to topspace X \<times> topspace Y)
   391            = (\<lambda>U. \<exists>S T. U = S \<times> T \<and> openin X S \<and> openin Y T)"
   392       (is "?lhs = ?rhs")
   393     proof -
   394       have "?rhs U" if "?lhs U" for U
   395       proof -
   396         have "topspace X \<times> topspace Y \<inter> \<Inter> T \<in> {A \<times> B |A B. A \<in> Collect (openin X) \<and> B \<in> Collect (openin Y)}"
   397           if "finite T" "T \<subseteq> \<X> \<union> \<Y>" for T
   398           using that
   399         proof induction
   400           case (insert B \<B>)
   401           then show ?case
   402             unfolding \<X>_def \<Y>_def
   403             apply (simp add: Int_ac subset_eq image_def)
   404             apply (metis (no_types) openin_Int openin_topspace Times_Int_Times)
   405             done
   406         qed auto
   407         then show ?thesis
   408           using that
   409           by (auto simp: subset_eq  elim!: relative_toE intersection_ofE)
   410       qed
   411       moreover
   412       have "?lhs Z" if Z: "?rhs Z" for Z
   413       proof -
   414         obtain U V where "Z = U \<times> V" "openin X U" "openin Y V"
   415           using Z by blast
   416         then have UV: "U \<times> V = (topspace X \<times> topspace Y) \<inter> (U \<times> V)"
   417           by (simp add: Sigma_mono inf_absorb2 openin_subset)
   418         moreover
   419         have "?lhs ((topspace X \<times> topspace Y) \<inter> (U \<times> V))"
   420         proof (rule relative_to_inc)
   421           show "(finite intersection_of (\<lambda>x. x \<in> \<X> \<or> x \<in> \<Y>)) (U \<times> V)"
   422             apply (simp add: intersection_of_def \<X>_def \<Y>_def)
   423             apply (rule_tac x="{(U \<times> topspace Y),(topspace X \<times> V)}" in exI)
   424             using \<open>openin X U\<close> \<open>openin Y V\<close> openin_subset UV apply (fastforce simp add:)
   425             done
   426         qed
   427         ultimately show ?thesis
   428           using \<open>Z = U \<times> V\<close> by auto
   429       qed
   430       ultimately show ?thesis
   431         by meson
   432     qed
   433     then show "topology (arbitrary union_of (finite intersection_of (\<lambda>x. x \<in> \<X> \<union> \<Y>)
   434            relative_to (topspace X \<times> topspace Y))) =
   435         prod_topology X Y"
   436       by (simp add: prod_topology_def)
   437   qed
   438   ultimately show ?thesis
   439     using False by blast
   440 qed
   441 
   442 lemma compactin_Times:
   443    "compactin (prod_topology X Y) (S \<times> T) \<longleftrightarrow> S = {} \<or> T = {} \<or> compactin X S \<and> compactin Y T"
   444   by (auto simp: compactin_subspace subtopology_Times compact_space_prod_topology)
   445 
   446 subsection\<open>Homeomorphic maps\<close>
   447 
   448 lemma homeomorphic_maps_prod:
   449    "homeomorphic_maps (prod_topology X Y) (prod_topology X' Y') (\<lambda>(x,y). (f x, g y)) (\<lambda>(x,y). (f' x, g' y)) \<longleftrightarrow>
   450         topspace(prod_topology X Y) = {} \<and>
   451         topspace(prod_topology X' Y') = {} \<or>
   452         homeomorphic_maps X X' f f' \<and>
   453         homeomorphic_maps Y Y' g g'"
   454   unfolding homeomorphic_maps_def continuous_map_prod_top
   455   by (auto simp: continuous_map_def homeomorphic_maps_def continuous_map_prod_top)
   456 
   457 lemma embedding_map_graph:
   458    "embedding_map X (prod_topology X Y) (\<lambda>x. (x, f x)) \<longleftrightarrow> continuous_map X Y f"
   459     (is "?lhs = ?rhs")
   460 proof
   461   assume L: ?lhs
   462   have "snd \<circ> (\<lambda>x. (x, f x)) = f"
   463     by force
   464   moreover have "continuous_map X Y (snd \<circ> (\<lambda>x. (x, f x)))"
   465     using L
   466     unfolding embedding_map_def
   467     by (meson continuous_map_in_subtopology continuous_map_snd_of homeomorphic_imp_continuous_map)
   468   ultimately show ?rhs
   469     by simp
   470 next
   471   assume R: ?rhs
   472   then show ?lhs
   473     unfolding homeomorphic_map_maps embedding_map_def homeomorphic_maps_def
   474     by (rule_tac x=fst in exI)
   475        (auto simp: continuous_map_in_subtopology continuous_map_paired continuous_map_from_subtopology
   476                    continuous_map_fst topspace_subtopology)
   477 qed
   478 
   479 lemma homeomorphic_space_prod_topology:
   480    "\<lbrakk>X homeomorphic_space X''; Y homeomorphic_space Y'\<rbrakk>
   481         \<Longrightarrow> prod_topology X Y homeomorphic_space prod_topology X'' Y'"
   482 using homeomorphic_maps_prod unfolding homeomorphic_space_def by blast
   483 
   484 lemma prod_topology_homeomorphic_space_left:
   485    "topspace Y = {b} \<Longrightarrow> prod_topology X Y homeomorphic_space X"
   486   unfolding homeomorphic_space_def
   487   by (rule_tac x=fst in exI) (simp add: homeomorphic_map_def inj_on_def flip: homeomorphic_map_maps)
   488 
   489 lemma prod_topology_homeomorphic_space_right:
   490    "topspace X = {a} \<Longrightarrow> prod_topology X Y homeomorphic_space Y"
   491   unfolding homeomorphic_space_def
   492   by (rule_tac x=snd in exI) (simp add: homeomorphic_map_def inj_on_def flip: homeomorphic_map_maps)
   493 
   494 
   495 lemma homeomorphic_space_prod_topology_sing1:
   496      "b \<in> topspace Y \<Longrightarrow> X homeomorphic_space (prod_topology X (subtopology Y {b}))"
   497   by (metis empty_subsetI homeomorphic_space_sym inf.absorb_iff2 insert_subset prod_topology_homeomorphic_space_left topspace_subtopology)
   498 
   499 lemma homeomorphic_space_prod_topology_sing2:
   500      "a \<in> topspace X \<Longrightarrow> Y homeomorphic_space (prod_topology (subtopology X {a}) Y)"
   501   by (metis empty_subsetI homeomorphic_space_sym inf.absorb_iff2 insert_subset prod_topology_homeomorphic_space_right topspace_subtopology)
   502 
   503 lemma topological_property_of_prod_component:
   504   assumes major: "P(prod_topology X Y)"
   505     and X: "\<And>x. \<lbrakk>x \<in> topspace X; P(prod_topology X Y)\<rbrakk> \<Longrightarrow> P(subtopology (prod_topology X Y) ({x} \<times> topspace Y))"
   506     and Y: "\<And>y. \<lbrakk>y \<in> topspace Y; P(prod_topology X Y)\<rbrakk> \<Longrightarrow> P(subtopology (prod_topology X Y) (topspace X \<times> {y}))"
   507     and PQ:  "\<And>X X'. X homeomorphic_space X' \<Longrightarrow> (P X \<longleftrightarrow> Q X')"
   508     and PR: "\<And>X X'. X homeomorphic_space X' \<Longrightarrow> (P X \<longleftrightarrow> R X')"
   509   shows "topspace(prod_topology X Y) = {} \<or> Q X \<and> R Y"
   510 proof -
   511   have "Q X \<and> R Y" if "topspace(prod_topology X Y) \<noteq> {}"
   512   proof -
   513     from that obtain a b where a: "a \<in> topspace X" and b: "b \<in> topspace Y"
   514       by force
   515     show ?thesis
   516       using X [OF a major] and Y [OF b major] homeomorphic_space_prod_topology_sing1 [OF b, of X] homeomorphic_space_prod_topology_sing2 [OF a, of Y]
   517       by (simp add: subtopology_Times) (meson PQ PR homeomorphic_space_prod_topology_sing2 homeomorphic_space_sym)
   518   qed
   519   then show ?thesis by metis
   520 qed
   521 
   522 lemma limitin_pairwise:
   523    "limitin (prod_topology X Y) f l F \<longleftrightarrow> limitin X (fst \<circ> f) (fst l) F \<and> limitin Y (snd \<circ> f) (snd l) F"
   524     (is "?lhs = ?rhs")
   525 proof
   526   assume ?lhs
   527   then obtain a b where ev: "\<And>U. \<lbrakk>(a,b) \<in> U; openin (prod_topology X Y) U\<rbrakk> \<Longrightarrow> \<forall>\<^sub>F x in F. f x \<in> U"
   528                         and a: "a \<in> topspace X" and b: "b \<in> topspace Y" and l: "l = (a,b)"
   529     by (auto simp: limitin_def)
   530   moreover have "\<forall>\<^sub>F x in F. fst (f x) \<in> U" if "openin X U" "a \<in> U" for U
   531   proof -
   532     have "\<forall>\<^sub>F c in F. f c \<in> U \<times> topspace Y"
   533       using b that ev [of "U \<times> topspace Y"] by (auto simp: openin_prod_topology_alt)
   534     then show ?thesis
   535       by (rule eventually_mono) (metis (mono_tags, lifting) SigmaE2 prod.collapse)
   536   qed
   537   moreover have "\<forall>\<^sub>F x in F. snd (f x) \<in> U" if "openin Y U" "b \<in> U" for U
   538   proof -
   539     have "\<forall>\<^sub>F c in F. f c \<in> topspace X \<times> U"
   540       using a that ev [of "topspace X \<times> U"] by (auto simp: openin_prod_topology_alt)
   541     then show ?thesis
   542       by (rule eventually_mono) (metis (mono_tags, lifting) SigmaE2 prod.collapse)
   543   qed
   544   ultimately show ?rhs
   545     by (simp add: limitin_def)
   546 next
   547   have "limitin (prod_topology X Y) f (a,b) F"
   548     if "limitin X (fst \<circ> f) a F" "limitin Y (snd \<circ> f) b F" for a b
   549     using that
   550   proof (clarsimp simp: limitin_def)
   551     fix Z :: "('a \<times> 'b) set"
   552     assume a: "a \<in> topspace X" "\<forall>U. openin X U \<and> a \<in> U \<longrightarrow> (\<forall>\<^sub>F x in F. fst (f x) \<in> U)"
   553       and b: "b \<in> topspace Y" "\<forall>U. openin Y U \<and> b \<in> U \<longrightarrow> (\<forall>\<^sub>F x in F. snd (f x) \<in> U)"
   554       and Z: "openin (prod_topology X Y) Z" "(a, b) \<in> Z"
   555     then obtain U V where "openin X U" "openin Y V" "a \<in> U" "b \<in> V" "U \<times> V \<subseteq> Z"
   556       using Z by (force simp: openin_prod_topology_alt)
   557     then have "\<forall>\<^sub>F x in F. fst (f x) \<in> U" "\<forall>\<^sub>F x in F. snd (f x) \<in> V"
   558       by (simp_all add: a b)
   559     then show "\<forall>\<^sub>F x in F. f x \<in> Z"
   560       by (rule eventually_elim2) (use \<open>U \<times> V \<subseteq> Z\<close> subsetD in auto)
   561   qed
   562   then show "?rhs \<Longrightarrow> ?lhs"
   563     by (metis prod.collapse)
   564 qed
   565 
   566 end
   567