section‹The binary product topology› theory Product_Topology imports Function_Topology begin section ‹Product Topology› subsection‹Definition› definition prod_topology :: "'a topology ⇒ 'b topology ⇒ ('a × 'b) topology" where "prod_topology X Y ≡ topology (arbitrary union_of (λU. U ∈ {S × T |S T. openin X S ∧ openin Y T}))" lemma open_product_open: assumes "open A" shows "∃𝒰. 𝒰 ⊆ {S × T |S T. open S ∧ open T} ∧ ⋃ 𝒰 = A" proof - obtain f g where *: "⋀u. u ∈ A ⟹ open (f u) ∧ open (g u) ∧ u ∈ (f u) × (g u) ∧ (f u) × (g u) ⊆ A" using open_prod_def [of A] assms by metis let ?𝒰 = "(λu. f u × g u) ` A" show ?thesis by (rule_tac x="?𝒰" in exI) (auto simp: dest: *) qed lemma open_product_open_eq: "(arbitrary union_of (λU. ∃S T. U = S × T ∧ open S ∧ open T)) = open" by (force simp: union_of_def arbitrary_def intro: open_product_open open_Times) lemma openin_prod_topology: "openin (prod_topology X Y) = arbitrary union_of (λU. U ∈ {S × T |S T. openin X S ∧ openin Y T})" unfolding prod_topology_def proof (rule topology_inverse') show "istopology (arbitrary union_of (λU. U ∈ {S × T |S T. openin X S ∧ openin Y T}))" apply (rule istopology_base, simp) by (metis openin_Int Times_Int_Times) qed lemma topspace_prod_topology [simp]: "topspace (prod_topology X Y) = topspace X × topspace Y" proof - have "topspace(prod_topology X Y) = ⋃ (Collect (openin (prod_topology X Y)))" (is "_ = ?Z") unfolding topspace_def .. also have "… = topspace X × topspace Y" proof show "?Z ⊆ topspace X × topspace Y" apply (auto simp: openin_prod_topology union_of_def arbitrary_def) using openin_subset by force+ next have *: "∃A B. topspace X × topspace Y = A × B ∧ openin X A ∧ openin Y B" by blast show "topspace X × topspace Y ⊆ ?Z" apply (rule Union_upper) using * by (simp add: openin_prod_topology arbitrary_union_of_inc) qed finally show ?thesis . qed lemma prod_topology_trivial_iff [simp]: "prod_topology X Y = trivial_topology ⟷ X = trivial_topology ∨ Y = trivial_topology" by (metis (full_types) Sigma_empty1 null_topspace_iff_trivial subset_empty times_subset_iff topspace_prod_topology) lemma subtopology_Times: shows "subtopology (prod_topology X Y) (S × T) = prod_topology (subtopology X S) (subtopology Y T)" proof - have "((λU. ∃S T. U = S × T ∧ openin X S ∧ openin Y T) relative_to S × T) = (λU. ∃S' T'. U = S' × T' ∧ (openin X relative_to S) S' ∧ (openin Y relative_to T) T')" by (auto simp: relative_to_def Times_Int_Times fun_eq_iff) metis then show ?thesis by (simp add: topology_eq openin_prod_topology arbitrary_union_of_relative_to flip: openin_relative_to) qed lemma prod_topology_subtopology: "prod_topology (subtopology X S) Y = subtopology (prod_topology X Y) (S × topspace Y)" "prod_topology X (subtopology Y T) = subtopology (prod_topology X Y) (topspace X × T)" by (auto simp: subtopology_Times) lemma prod_topology_discrete_topology: "discrete_topology (S × T) = prod_topology (discrete_topology S) (discrete_topology T)" by (auto simp: discrete_topology_unique openin_prod_topology intro: arbitrary_union_of_inc) lemma prod_topology_euclidean [simp]: "prod_topology euclidean euclidean = euclidean" by (simp add: prod_topology_def open_product_open_eq) lemma prod_topology_subtopology_eu [simp]: "prod_topology (subtopology euclidean S) (subtopology euclidean T) = subtopology euclidean (S × T)" by (simp add: prod_topology_subtopology subtopology_subtopology Times_Int_Times) lemma openin_prod_topology_alt: "openin (prod_topology X Y) S ⟷ (∀x y. (x,y) ∈ S ⟶ (∃U V. openin X U ∧ openin Y V ∧ x ∈ U ∧ y ∈ V ∧ U × V ⊆ S))" apply (auto simp: openin_prod_topology arbitrary_union_of_alt, fastforce) by (metis mem_Sigma_iff) lemma open_map_fst: "open_map (prod_topology X Y) X fst" unfolding open_map_def openin_prod_topology_alt by (force simp: openin_subopen [of X "fst ` _"] intro: subset_fst_imageI) lemma open_map_snd: "open_map (prod_topology X Y) Y snd" unfolding open_map_def openin_prod_topology_alt by (force simp: openin_subopen [of Y "snd ` _"] intro: subset_snd_imageI) lemma openin_prod_Times_iff: "openin (prod_topology X Y) (S × T) ⟷ S = {} ∨ T = {} ∨ openin X S ∧ openin Y T" proof (cases "S = {} ∨ T = {}") case False then show ?thesis apply (simp add: openin_prod_topology_alt openin_subopen [of X S] openin_subopen [of Y T] times_subset_iff, safe) apply (meson|force)+ done qed force lemma closure_of_Times: "(prod_topology X Y) closure_of (S × T) = (X closure_of S) × (Y closure_of T)" (is "?lhs = ?rhs") proof show "?lhs ⊆ ?rhs" by (clarsimp simp: closure_of_def openin_prod_topology_alt) blast show "?rhs ⊆ ?lhs" by (clarsimp simp: closure_of_def openin_prod_topology_alt) (meson SigmaI subsetD) qed lemma closedin_prod_Times_iff: "closedin (prod_topology X Y) (S × T) ⟷ S = {} ∨ T = {} ∨ closedin X S ∧ closedin Y T" by (auto simp: closure_of_Times times_eq_iff simp flip: closure_of_eq) lemma interior_of_Times: "(prod_topology X Y) interior_of (S × T) = (X interior_of S) × (Y interior_of T)" proof (rule interior_of_unique) show "(X interior_of S) × Y interior_of T ⊆ S × T" by (simp add: Sigma_mono interior_of_subset) show "openin (prod_topology X Y) ((X interior_of S) × Y interior_of T)" by (simp add: openin_prod_Times_iff) next show "T' ⊆ (X interior_of S) × Y interior_of T" if "T' ⊆ S × T" "openin (prod_topology X Y) T'" for T' proof (clarsimp; intro conjI) fix a :: "'a" and b :: "'b" assume "(a, b) ∈ T'" with that obtain U V where UV: "openin X U" "openin Y V" "a ∈ U" "b ∈ V" "U × V ⊆ T'" by (metis openin_prod_topology_alt) then show "a ∈ X interior_of S" using interior_of_maximal_eq that(1) by fastforce show "b ∈ Y interior_of T" using UV interior_of_maximal_eq that(1) by (metis SigmaI mem_Sigma_iff subset_eq) qed qed text ‹Missing the opposite direction. Does it hold? A converse is proved for proper maps, a stronger condition› lemma closed_map_prod: assumes "closed_map (prod_topology X Y) (prod_topology X' Y') (λ(x,y). (f x, g y))" shows "(prod_topology X Y) = trivial_topology ∨ closed_map X X' f ∧ closed_map Y Y' g" proof (cases "(prod_topology X Y) = trivial_topology") case False then have ne: "topspace X ≠ {}" "topspace Y ≠ {}" by (auto simp flip: null_topspace_iff_trivial) have "closed_map X X' f" unfolding closed_map_def proof (intro strip) fix C assume "closedin X C" show "closedin X' (f ` C)" proof (cases "C={}") case False with assms have "closedin (prod_topology X' Y') ((λ(x,y). (f x, g y)) ` (C × topspace Y))" by (simp add: ‹closedin X C› closed_map_def closedin_prod_Times_iff) with False ne show ?thesis by (simp add: image_paired_Times closedin_Times closedin_prod_Times_iff) qed auto qed moreover have "closed_map Y Y' g" unfolding closed_map_def proof (intro strip) fix C assume "closedin Y C" show "closedin Y' (g ` C)" proof (cases "C={}") case False with assms have "closedin (prod_topology X' Y') ((λ(x,y). (f x, g y)) ` (topspace X × C))" by (simp add: ‹closedin Y C› closed_map_def closedin_prod_Times_iff) with False ne show ?thesis by (simp add: image_paired_Times closedin_Times closedin_prod_Times_iff) qed auto qed ultimately show ?thesis by (auto simp: False) qed auto subsection ‹Continuity› lemma continuous_map_pairwise: "continuous_map Z (prod_topology X Y) f ⟷ continuous_map Z X (fst ∘ f) ∧ continuous_map Z Y (snd ∘ f)" (is "?lhs = ?rhs") proof - let ?g = "fst ∘ f" and ?h = "snd ∘ f" have f: "f x = (?g x, ?h x)" for x by auto show ?thesis proof (cases "?g ∈ topspace Z → topspace X ∧ ?h ∈ topspace Z → topspace Y") case True show ?thesis proof safe assume "continuous_map Z (prod_topology X Y) f" then have "openin Z {x ∈ topspace Z. fst (f x) ∈ U}" if "openin X U" for U unfolding continuous_map_def using True that apply clarify apply (drule_tac x="U × topspace Y" in spec) by (auto simp: openin_prod_Times_iff mem_Times_iff Pi_iff cong: conj_cong) with True show "continuous_map Z X (fst ∘ f)" by (auto simp: continuous_map_def) next assume "continuous_map Z (prod_topology X Y) f" then have "openin Z {x ∈ topspace Z. snd (f x) ∈ V}" if "openin Y V" for V unfolding continuous_map_def using True that apply clarify apply (drule_tac x="topspace X × V" in spec) by (simp add: openin_prod_Times_iff mem_Times_iff Pi_iff cong: conj_cong) with True show "continuous_map Z Y (snd ∘ f)" by (auto simp: continuous_map_def) next assume Z: "continuous_map Z X (fst ∘ f)" "continuous_map Z Y (snd ∘ f)" have *: "openin Z {x ∈ topspace Z. f x ∈ W}" if "⋀w. w ∈ W ⟹ ∃U V. openin X U ∧ openin Y V ∧ w ∈ U × V ∧ U × V ⊆ W" for W proof (subst openin_subopen, clarify) fix x :: "'a" assume "x ∈ topspace Z" and "f x ∈ W" with that [OF ‹f x ∈ W›] obtain U V where UV: "openin X U" "openin Y V" "f x ∈ U × V" "U × V ⊆ W" by auto with Z UV show "∃T. openin Z T ∧ x ∈ T ∧ T ⊆ {x ∈ topspace Z. f x ∈ W}" apply (rule_tac x="{x ∈ topspace Z. ?g x ∈ U} ∩ {x ∈ topspace Z. ?h x ∈ V}" in exI) apply (auto simp: ‹x ∈ topspace Z› continuous_map_def) done qed show "continuous_map Z (prod_topology X Y) f" using True by (force simp: continuous_map_def openin_prod_topology_alt mem_Times_iff *) qed qed (force simp: continuous_map_def) qed lemma continuous_map_paired: "continuous_map Z (prod_topology X Y) (λx. (f x,g x)) ⟷ continuous_map Z X f ∧ continuous_map Z Y g" by (simp add: continuous_map_pairwise o_def) lemma continuous_map_pairedI [continuous_intros]: "⟦continuous_map Z X f; continuous_map Z Y g⟧ ⟹ continuous_map Z (prod_topology X Y) (λx. (f x,g x))" by (simp add: continuous_map_pairwise o_def) lemma continuous_map_fst [continuous_intros]: "continuous_map (prod_topology X Y) X fst" using continuous_map_pairwise [of "prod_topology X Y" X Y id] by (simp add: continuous_map_pairwise) lemma continuous_map_snd [continuous_intros]: "continuous_map (prod_topology X Y) Y snd" using continuous_map_pairwise [of "prod_topology X Y" X Y id] by (simp add: continuous_map_pairwise) lemma continuous_map_fst_of [continuous_intros]: "continuous_map Z (prod_topology X Y) f ⟹ continuous_map Z X (fst ∘ f)" by (simp add: continuous_map_pairwise) lemma continuous_map_snd_of [continuous_intros]: "continuous_map Z (prod_topology X Y) f ⟹ continuous_map Z Y (snd ∘ f)" by (simp add: continuous_map_pairwise) lemma continuous_map_prod_fst: "i ∈ I ⟹ continuous_map (prod_topology (product_topology (λi. Y) I) X) Y (λx. fst x i)" using continuous_map_componentwise_UNIV continuous_map_fst by fastforce lemma continuous_map_prod_snd: "i ∈ I ⟹ continuous_map (prod_topology X (product_topology (λi. Y) I)) Y (λx. snd x i)" using continuous_map_componentwise_UNIV continuous_map_snd by fastforce lemma continuous_map_if_iff [simp]: "continuous_map X Y (λx. if P then f x else g x) ⟷ continuous_map X Y (if P then f else g)" by simp lemma continuous_map_if [continuous_intros]: "⟦P ⟹ continuous_map X Y f; ~P ⟹ continuous_map X Y g⟧ ⟹ continuous_map X Y (λx. if P then f x else g x)" by simp lemma prod_topology_trivial1 [simp]: "prod_topology trivial_topology Y = trivial_topology" using continuous_map_fst continuous_map_on_empty2 by blast lemma prod_topology_trivial2 [simp]: "prod_topology X trivial_topology = trivial_topology" using continuous_map_snd continuous_map_on_empty2 by blast lemma continuous_map_subtopology_fst [continuous_intros]: "continuous_map (subtopology (prod_topology X Y) Z) X fst" using continuous_map_from_subtopology continuous_map_fst by force lemma continuous_map_subtopology_snd [continuous_intros]: "continuous_map (subtopology (prod_topology X Y) Z) Y snd" using continuous_map_from_subtopology continuous_map_snd by force lemma quotient_map_fst [simp]: "quotient_map(prod_topology X Y) X fst ⟷ (Y = trivial_topology ⟶ X = trivial_topology)" apply (simp add: continuous_open_quotient_map open_map_fst continuous_map_fst) by (metis null_topspace_iff_trivial) lemma quotient_map_snd [simp]: "quotient_map(prod_topology X Y) Y snd ⟷ (X = trivial_topology ⟶ Y = trivial_topology)" apply (simp add: continuous_open_quotient_map open_map_snd continuous_map_snd) by (metis null_topspace_iff_trivial) lemma retraction_map_fst: "retraction_map (prod_topology X Y) X fst ⟷ (Y = trivial_topology ⟶ X = trivial_topology)" proof (cases "Y = trivial_topology") case True then show ?thesis using continuous_map_image_subset_topspace by (auto simp: retraction_map_def retraction_maps_def continuous_map_pairwise) next case False have "∃g. continuous_map X (prod_topology X Y) g ∧ (∀x∈topspace X. fst (g x) = x)" if y: "y ∈ topspace Y" for y by (rule_tac x="λx. (x,y)" in exI) (auto simp: y continuous_map_paired) with False have "retraction_map (prod_topology X Y) X fst" by (fastforce simp: retraction_map_def retraction_maps_def continuous_map_fst) with False show ?thesis by simp qed lemma retraction_map_snd: "retraction_map (prod_topology X Y) Y snd ⟷ (X = trivial_topology ⟶ Y = trivial_topology)" proof (cases "X = trivial_topology") case True then show ?thesis using continuous_map_image_subset_topspace by (fastforce simp: retraction_map_def retraction_maps_def continuous_map_fst) next case False have "∃g. continuous_map Y (prod_topology X Y) g ∧ (∀y∈topspace Y. snd (g y) = y)" if x: "x ∈ topspace X" for x by (rule_tac x="λy. (x,y)" in exI) (auto simp: x continuous_map_paired) with False have "retraction_map (prod_topology X Y) Y snd" by (fastforce simp: retraction_map_def retraction_maps_def continuous_map_snd) with False show ?thesis by simp qed lemma continuous_map_of_fst: "continuous_map (prod_topology X Y) Z (f ∘ fst) ⟷ Y = trivial_topology ∨ continuous_map X Z f" proof (cases "Y = trivial_topology") case True then show ?thesis by (simp add: continuous_map_on_empty) next case False then show ?thesis by (simp add: continuous_compose_quotient_map_eq) qed lemma continuous_map_of_snd: "continuous_map (prod_topology X Y) Z (f ∘ snd) ⟷ X = trivial_topology ∨ continuous_map Y Z f" proof (cases "X = trivial_topology") case True then show ?thesis by (simp add: continuous_map_on_empty) next case False then show ?thesis by (simp add: continuous_compose_quotient_map_eq) qed lemma continuous_map_prod_top: "continuous_map (prod_topology X Y) (prod_topology X' Y') (λ(x,y). (f x, g y)) ⟷ (prod_topology X Y) = trivial_topology ∨ continuous_map X X' f ∧ continuous_map Y Y' g" proof (cases "(prod_topology X Y) = trivial_topology") case False then show ?thesis by (auto simp: continuous_map_paired case_prod_unfold continuous_map_of_fst [unfolded o_def] continuous_map_of_snd [unfolded o_def]) qed auto lemma in_prod_topology_closure_of: assumes "z ∈ (prod_topology X Y) closure_of S" shows "fst z ∈ X closure_of (fst ` S)" "snd z ∈ Y closure_of (snd ` S)" using assms continuous_map_eq_image_closure_subset continuous_map_fst apply fastforce using assms continuous_map_eq_image_closure_subset continuous_map_snd apply fastforce done proposition compact_space_prod_topology: "compact_space(prod_topology X Y) ⟷ (prod_topology X Y) = trivial_topology ∨ compact_space X ∧ compact_space Y" proof (cases "(prod_topology X Y) = trivial_topology") case True then show ?thesis by fastforce next case False then have non_mt: "topspace X ≠ {}" "topspace Y ≠ {}" by auto have "compact_space X" "compact_space Y" if "compact_space(prod_topology X Y)" proof - have "compactin X (fst ` (topspace X × topspace Y))" by (metis compact_space_def continuous_map_fst image_compactin that topspace_prod_topology) moreover have "compactin Y (snd ` (topspace X × topspace Y))" by (metis compact_space_def continuous_map_snd image_compactin that topspace_prod_topology) ultimately show "compact_space X" "compact_space Y" using non_mt by (auto simp: compact_space_def) qed moreover define 𝒳 where "𝒳 ≡ (λV. topspace X × V) ` Collect (openin Y)" define 𝒴 where "𝒴 ≡ (λU. U × topspace Y) ` Collect (openin X)" have "compact_space(prod_topology X Y)" if "compact_space X" "compact_space Y" proof (rule Alexander_subbase_alt) show toptop: "topspace X × topspace Y ⊆ ⋃(𝒳 ∪ 𝒴)" unfolding 𝒳_def 𝒴_def by auto fix 𝒞 :: "('a × 'b) set set" assume 𝒞: "𝒞 ⊆ 𝒳 ∪ 𝒴" "topspace X × topspace Y ⊆ ⋃𝒞" then obtain 𝒳' 𝒴' where XY: "𝒳' ⊆ 𝒳" "𝒴' ⊆ 𝒴" and 𝒞eq: "𝒞 = 𝒳' ∪ 𝒴'" using subset_UnE by metis then have sub: "topspace X × topspace Y ⊆ ⋃(𝒳' ∪ 𝒴')" using 𝒞 by simp obtain 𝒰 𝒱 where 𝒰: "⋀U. U ∈ 𝒰 ⟹ openin X U" "𝒴' = (λU. U × topspace Y) ` 𝒰" and 𝒱: "⋀V. V ∈ 𝒱 ⟹ openin Y V" "𝒳' = (λV. topspace X × V) ` 𝒱" using XY by (clarsimp simp add: 𝒳_def 𝒴_def subset_image_iff) (force simp: subset_iff) have "∃𝒟. finite 𝒟 ∧ 𝒟 ⊆ 𝒳' ∪ 𝒴' ∧ topspace X × topspace Y ⊆ ⋃ 𝒟" proof - have "topspace X ⊆ ⋃𝒰 ∨ topspace Y ⊆ ⋃𝒱" using 𝒰 𝒱 𝒞 𝒞eq by auto then have *: "∃𝒟. finite 𝒟 ∧ (∀x ∈ 𝒟. x ∈ (λV. topspace X × V) ` 𝒱 ∨ x ∈ (λU. U × topspace Y) ` 𝒰) ∧ (topspace X × topspace Y ⊆ ⋃𝒟)" proof assume "topspace X ⊆ ⋃𝒰" with ‹compact_space X› 𝒰 obtain ℱ where "finite ℱ" "ℱ ⊆ 𝒰" "topspace X ⊆ ⋃ℱ" by (meson compact_space_alt) with that show ?thesis by (rule_tac x="(λD. D × topspace Y) ` ℱ" in exI) auto next assume "topspace Y ⊆ ⋃𝒱" with ‹compact_space Y› 𝒱 obtain ℱ where "finite ℱ" "ℱ ⊆ 𝒱" "topspace Y ⊆ ⋃ℱ" by (meson compact_space_alt) with that show ?thesis by (rule_tac x="(λC. topspace X × C) ` ℱ" in exI) auto qed then show ?thesis using that 𝒰 𝒱 by blast qed then show "∃𝒟. finite 𝒟 ∧ 𝒟 ⊆ 𝒞 ∧ topspace X × topspace Y ⊆ ⋃ 𝒟" using 𝒞 𝒞eq by blast next have "(finite intersection_of (λx. x ∈ 𝒳 ∨ x ∈ 𝒴) relative_to topspace X × topspace Y) = (λU. ∃S T. U = S × T ∧ openin X S ∧ openin Y T)" (is "?lhs = ?rhs") proof - have "?rhs U" if "?lhs U" for U proof - have "topspace X × topspace Y ∩ ⋂ T ∈ {A × B |A B. A ∈ Collect (openin X) ∧ B ∈ Collect (openin Y)}" if "finite T" "T ⊆ 𝒳 ∪ 𝒴" for T using that proof induction case (insert B ℬ) then show ?case unfolding 𝒳_def 𝒴_def apply (simp add: Int_ac subset_eq image_def) apply (metis (no_types) openin_Int openin_topspace Times_Int_Times) done qed auto then show ?thesis using that by (auto simp: subset_eq elim!: relative_toE intersection_ofE) qed moreover have "?lhs Z" if Z: "?rhs Z" for Z proof - obtain U V where "Z = U × V" "openin X U" "openin Y V" using Z by blast then have UV: "U × V = (topspace X × topspace Y) ∩ (U × V)" by (simp add: Sigma_mono inf_absorb2 openin_subset) moreover have "?lhs ((topspace X × topspace Y) ∩ (U × V))" proof (rule relative_to_inc) show "(finite intersection_of (λx. x ∈ 𝒳 ∨ x ∈ 𝒴)) (U × V)" apply (simp add: intersection_of_def 𝒳_def 𝒴_def) apply (rule_tac x="{(U × topspace Y),(topspace X × V)}" in exI) using ‹openin X U› ‹openin Y V› openin_subset UV apply (fastforce simp:) done qed ultimately show ?thesis using ‹Z = U × V› by auto qed ultimately show ?thesis by meson qed then show "topology (arbitrary union_of (finite intersection_of (λx. x ∈ 𝒳 ∪ 𝒴) relative_to (topspace X × topspace Y))) = prod_topology X Y" by (simp add: prod_topology_def) qed ultimately show ?thesis using False by blast qed lemma compactin_Times: "compactin (prod_topology X Y) (S × T) ⟷ S = {} ∨ T = {} ∨ compactin X S ∧ compactin Y T" by (auto simp: compactin_subspace subtopology_Times compact_space_prod_topology subtopology_trivial_iff) subsection‹Homeomorphic maps› lemma homeomorphic_maps_prod: "homeomorphic_maps (prod_topology X Y) (prod_topology X' Y') (λ(x,y). (f x, g y)) (λ(x,y). (f' x, g' y)) ⟷ (prod_topology X Y) = trivial_topology ∧ (prod_topology X' Y') = trivial_topology ∨ homeomorphic_maps X X' f f' ∧ homeomorphic_maps Y Y' g g'" (is "?lhs = ?rhs") proof show "?lhs ⟹ ?rhs" by (fastforce simp: homeomorphic_maps_def continuous_map_prod_top ball_conj_distrib) next show "?rhs ⟹ ?lhs" by (auto simp: homeomorphic_maps_def continuous_map_prod_top) qed lemma homeomorphic_maps_swap: "homeomorphic_maps (prod_topology X Y) (prod_topology Y X) (λ(x,y). (y,x)) (λ(y,x). (x,y))" by (auto simp: homeomorphic_maps_def case_prod_unfold continuous_map_fst continuous_map_pairedI continuous_map_snd) lemma homeomorphic_map_swap: "homeomorphic_map (prod_topology X Y) (prod_topology Y X) (λ(x,y). (y,x))" using homeomorphic_map_maps homeomorphic_maps_swap by metis lemma homeomorphic_space_prod_topology_swap: "(prod_topology X Y) homeomorphic_space (prod_topology Y X)" using homeomorphic_map_swap homeomorphic_space by blast lemma embedding_map_graph: "embedding_map X (prod_topology X Y) (λx. (x, f x)) ⟷ continuous_map X Y f" (is "?lhs = ?rhs") proof assume L: ?lhs have "snd ∘ (λx. (x, f x)) = f" by force moreover have "continuous_map X Y (snd ∘ (λx. (x, f x)))" using L unfolding embedding_map_def by (meson continuous_map_in_subtopology continuous_map_snd_of homeomorphic_imp_continuous_map) ultimately show ?rhs by simp next assume R: ?rhs then show ?lhs unfolding homeomorphic_map_maps embedding_map_def homeomorphic_maps_def by (rule_tac x=fst in exI) (auto simp: continuous_map_in_subtopology continuous_map_paired continuous_map_from_subtopology continuous_map_fst) qed lemma homeomorphic_space_prod_topology: "⟦X homeomorphic_space X''; Y homeomorphic_space Y'⟧ ⟹ prod_topology X Y homeomorphic_space prod_topology X'' Y'" using homeomorphic_maps_prod unfolding homeomorphic_space_def by blast lemma prod_topology_homeomorphic_space_left: "Y = discrete_topology {b} ⟹ prod_topology X Y homeomorphic_space X" unfolding homeomorphic_space_def apply (rule_tac x=fst in exI) apply (simp add: homeomorphic_map_def inj_on_def discrete_topology_unique flip: homeomorphic_map_maps) done lemma prod_topology_homeomorphic_space_right: "X = discrete_topology {a} ⟹ prod_topology X Y homeomorphic_space Y" unfolding homeomorphic_space_def by (meson homeomorphic_space_def homeomorphic_space_prod_topology_swap homeomorphic_space_trans prod_topology_homeomorphic_space_left) lemma homeomorphic_space_prod_topology_sing1: "b ∈ topspace Y ⟹ X homeomorphic_space (prod_topology X (subtopology Y {b}))" by (metis empty_subsetI homeomorphic_space_sym insert_subset prod_topology_homeomorphic_space_left subtopology_eq_discrete_topology_sing topspace_subtopology_subset) lemma homeomorphic_space_prod_topology_sing2: "a ∈ topspace X ⟹ Y homeomorphic_space (prod_topology (subtopology X {a}) Y)" by (metis empty_subsetI homeomorphic_space_sym insert_subset prod_topology_homeomorphic_space_right subtopology_eq_discrete_topology_sing topspace_subtopology_subset) lemma topological_property_of_prod_component: assumes major: "P(prod_topology X Y)" and X: "⋀x. ⟦x ∈ topspace X; P(prod_topology X Y)⟧ ⟹ P(subtopology (prod_topology X Y) ({x} × topspace Y))" and Y: "⋀y. ⟦y ∈ topspace Y; P(prod_topology X Y)⟧ ⟹ P(subtopology (prod_topology X Y) (topspace X × {y}))" and PQ: "⋀X X'. X homeomorphic_space X' ⟹ (P X ⟷ Q X')" and PR: "⋀X X'. X homeomorphic_space X' ⟹ (P X ⟷ R X')" shows "(prod_topology X Y) = trivial_topology ∨ Q X ∧ R Y" proof - have "Q X ∧ R Y" if "topspace(prod_topology X Y) ≠ {}" proof - from that obtain a b where a: "a ∈ topspace X" and b: "b ∈ topspace Y" by force show ?thesis 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] by (simp add: subtopology_Times) (meson PQ PR homeomorphic_space_prod_topology_sing2 homeomorphic_space_sym) qed then show ?thesis by force qed lemma limitin_pairwise: "limitin (prod_topology X Y) f l F ⟷ limitin X (fst ∘ f) (fst l) F ∧ limitin Y (snd ∘ f) (snd l) F" (is "?lhs = ?rhs") proof assume ?lhs then obtain a b where ev: "⋀U. ⟦(a,b) ∈ U; openin (prod_topology X Y) U⟧ ⟹ ∀⇩_{F}x in F. f x ∈ U" and a: "a ∈ topspace X" and b: "b ∈ topspace Y" and l: "l = (a,b)" by (auto simp: limitin_def) moreover have "∀⇩_{F}x in F. fst (f x) ∈ U" if "openin X U" "a ∈ U" for U proof - have "∀⇩_{F}c in F. f c ∈ U × topspace Y" using b that ev [of "U × topspace Y"] by (auto simp: openin_prod_topology_alt) then show ?thesis by (rule eventually_mono) (metis (mono_tags, lifting) SigmaE2 prod.collapse) qed moreover have "∀⇩_{F}x in F. snd (f x) ∈ U" if "openin Y U" "b ∈ U" for U proof - have "∀⇩_{F}c in F. f c ∈ topspace X × U" using a that ev [of "topspace X × U"] by (auto simp: openin_prod_topology_alt) then show ?thesis by (rule eventually_mono) (metis (mono_tags, lifting) SigmaE2 prod.collapse) qed ultimately show ?rhs by (simp add: limitin_def) next have "limitin (prod_topology X Y) f (a,b) F" if "limitin X (fst ∘ f) a F" "limitin Y (snd ∘ f) b F" for a b using that proof (clarsimp simp: limitin_def) fix Z :: "('a × 'b) set" assume a: "a ∈ topspace X" "∀U. openin X U ∧ a ∈ U ⟶ (∀⇩_{F}x in F. fst (f x) ∈ U)" and b: "b ∈ topspace Y" "∀U. openin Y U ∧ b ∈ U ⟶ (∀⇩_{F}x in F. snd (f x) ∈ U)" and Z: "openin (prod_topology X Y) Z" "(a, b) ∈ Z" then obtain U V where "openin X U" "openin Y V" "a ∈ U" "b ∈ V" "U × V ⊆ Z" using Z by (force simp: openin_prod_topology_alt) then have "∀⇩_{F}x in F. fst (f x) ∈ U" "∀⇩_{F}x in F. snd (f x) ∈ V" by (simp_all add: a b) then show "∀⇩_{F}x in F. f x ∈ Z" by (rule eventually_elim2) (use ‹U × V ⊆ Z› subsetD in auto) qed then show "?rhs ⟹ ?lhs" by (metis prod.collapse) qed proposition connected_space_prod_topology: "connected_space(prod_topology X Y) ⟷ (prod_topology X Y) = trivial_topology ∨ connected_space X ∧ connected_space Y" (is "?lhs=?rhs") proof (cases "(prod_topology X Y) = trivial_topology") case True then show ?thesis by auto next case False then have nonempty: "topspace X ≠ {}" "topspace Y ≠ {}" by force+ show ?thesis proof assume ?lhs then show ?rhs by (metis connected_space_quotient_map_image nonempty quotient_map_fst quotient_map_snd subtopology_eq_discrete_topology_empty) next assume ?rhs then have conX: "connected_space X" and conY: "connected_space Y" using False by blast+ have False if "openin (prod_topology X Y) U" and "openin (prod_topology X Y) V" and UV: "topspace X × topspace Y ⊆ U ∪ V" "U ∩ V = {}" and "U ≠ {}" and "V ≠ {}" for U V proof - have Usub: "U ⊆ topspace X × topspace Y" and Vsub: "V ⊆ topspace X × topspace Y" using that by (metis openin_subset topspace_prod_topology)+ obtain a b where ab: "(a,b) ∈ U" and a: "a ∈ topspace X" and b: "b ∈ topspace Y" using ‹U ≠ {}› Usub by auto have "¬ topspace X × topspace Y ⊆ U" using Usub Vsub ‹U ∩ V = {}› ‹V ≠ {}› by auto then obtain x y where x: "x ∈ topspace X" and y: "y ∈ topspace Y" and "(x,y) ∉ U" by blast have oX: "openin X {x ∈ topspace X. (x,y) ∈ U}" "openin X {x ∈ topspace X. (x,y) ∈ V}" and oY: "openin Y {y ∈ topspace Y. (a,y) ∈ U}" "openin Y {y ∈ topspace Y. (a,y) ∈ V}" by (force intro: openin_continuous_map_preimage [where Y = "prod_topology X Y"] simp: that continuous_map_pairwise o_def x y a)+ have 1: "topspace Y ⊆ {y ∈ topspace Y. (a,y) ∈ U} ∪ {y ∈ topspace Y. (a,y) ∈ V}" using a that(3) by auto have 2: "{y ∈ topspace Y. (a,y) ∈ U} ∩ {y ∈ topspace Y. (a,y) ∈ V} = {}" using that(4) by auto have 3: "{y ∈ topspace Y. (a,y) ∈ U} ≠ {}" using ab b by auto have 4: "{y ∈ topspace Y. (a,y) ∈ V} ≠ {}" proof - show ?thesis using connected_spaceD [OF conX oX] UV ‹(x,y) ∉ U› a x y disjoint_iff_not_equal by blast qed show ?thesis using connected_spaceD [OF conY oY 1 2 3 4] by auto qed then show ?lhs unfolding connected_space_def topspace_prod_topology by blast qed qed lemma connectedin_Times: "connectedin (prod_topology X Y) (S × T) ⟷ S = {} ∨ T = {} ∨ connectedin X S ∧ connectedin Y T" by (auto simp: connectedin_def subtopology_Times connected_space_prod_topology subtopology_trivial_iff) end