(* This file is a part of IsarMathLib - a library of formalized mathematics for Isabelle/Isar. Copyright (C) 2005 - 2022 Slawomir Kolodynski This program is free software Redistribution and use in source and binary forms, with or without modification, are permitted provided that the following conditions are met: 1. Redistributions of source code must retain the above copyright notice, this list of conditions and the following disclaimer. 2. Redistributions in binary form must reproduce the above copyright notice, this list of conditions and the following disclaimer in the documentation and/or other materials provided with the distribution. 3. The name of the author may not be used to endorse or promote products derived from this software without specific prior written permission. THIS SOFTWARE IS PROVIDED BY THE AUTHOR ``AS IS'' AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES LOSS OF USE, DATA, OR PROFITS OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. *) section ‹Topology 2› theory Topology_ZF_2 imports Topology_ZF_1 func1 Fol1 begin text‹This theory continues the series on general topology and covers the definition and basic properties of continuous functions. We also introduce the notion of homeomorphism an prove the pasting lemma.› subsection‹Continuous functions.› text‹In this section we define continuous functions and prove that certain conditions are equivalent to a function being continuous.› text‹In standard math we say that a function is continuous with respect to two topologies $\tau_1 ,\tau_2 $ if the inverse image of sets from topology $\tau_2$ are in $\tau_1$. Here we define a predicate that is supposed to reflect that definition, with a difference that we don't require in the definition that $\tau_1 ,\tau_2 $ are topologies. This means for example that when we define measurable functions, the definition will be the same. The notation ‹f-``(A)› means the inverse image of (a set) $A$ with respect to (a function) $f$. › definition "IsContinuous(τ⇩_{1},τ⇩_{2},f) ≡ (∀U∈τ⇩_{2}. f-``(U) ∈ τ⇩_{1})" text‹The space of continuous functions mapping $X=\bigcup \tau_1$ to $Y=\bigcup \tau_2$ will be denoted ‹Cont(τ⇩_{1},τ⇩_{2})›. › definition "Cont(τ⇩_{1},τ⇩_{2}) ≡ {f∈(⋃τ⇩_{1})→(⋃τ⇩_{2}). IsContinuous(τ⇩_{1},τ⇩_{2},f)}" text‹A trivial example of a continuous function - identity is continuous.› lemma id_cont: shows "IsContinuous(τ,τ,id(⋃τ))" proof - { fix U assume "U∈τ" then have "id(⋃τ)-``(U) = U" using vimage_id_same by auto with ‹U∈τ› have "id(⋃τ)-``(U) ∈ τ" by simp } then show "IsContinuous(τ,τ,id(⋃τ))" unfolding IsContinuous_def by simp qed text‹Identity is in the space of continuous functions from $\bigcup \tau$ to itself.› lemma id_cont_sp: shows "{⟨x,x⟩. x∈⋃τ} ∈ Cont(τ,τ)" proof - have "id(⋃τ) : ⋃τ → ⋃τ" and "IsContinuous(τ,τ,id(⋃τ))" using id_type id_cont by auto moreover have "id(⋃τ) = {⟨x,x⟩. x∈⋃τ}" by blast ultimately show ?thesis unfolding Cont_def by simp qed text‹A constant function is continuous.› lemma const_cont: assumes "T {is a topology}" shows "IsContinuous(T,τ,ConstantFunction(⋃T,c))" proof - let ?C = "ConstantFunction(⋃T,c)" { fix U assume "U∈τ" have "?C-``(U) ∈ T" proof - { assume "c∈U" with assms have "?C-``(U) ∈ T" using carr_open const_vimage_domain by simp } moreover { assume "c∉U" with assms have "?C-``(U) ∈ T" using empty_open const_vimage_empty by simp } ultimately show "?C-``(U) ∈ T" by auto qed } then show ?thesis unfolding IsContinuous_def by simp qed text‹If $c\in Y = \bigcup S$, then the constant function defined on $X=\bigcup T $ that is equal to $c$ is in the the space of continuous functions from $X$ to $Y$. › lemma const_cont_sp: assumes "T {is a topology}" "c∈⋃S" shows "{⟨x,c⟩. x∈⋃T} ∈ Cont(T,S)" using assms ZF_fun_from_total const_fun_def_alt const_cont unfolding Cont_def by simp text‹We will work with a pair of topological spaces. The following locale sets up our context that consists of two topologies $\tau_1,\tau_2$ and a continuous function $f: X_1 \rightarrow X_2$, where $X_i$ is defined as $\bigcup\tau_i$ for $i=1,2$. We also define notation ‹cl⇩_{1}(A)› and ‹cl⇩_{2}(A)› for closure of a set $A$ in topologies $\tau_1$ and $\tau_2$, respectively.› locale two_top_spaces0 = fixes τ⇩_{1}assumes tau1_is_top: "τ⇩_{1}{is a topology}" fixes τ⇩_{2}assumes tau2_is_top: "τ⇩_{2}{is a topology}" fixes X⇩_{1}defines X1_def [simp]: "X⇩_{1}≡ ⋃τ⇩_{1}" fixes X⇩_{2}defines X2_def [simp]: "X⇩_{2}≡ ⋃τ⇩_{2}" fixes f assumes fmapAssum: "f: X⇩_{1}→ X⇩_{2}" fixes isContinuous ("_ {is continuous}" [50] 50) defines isContinuous_def [simp]: "g {is continuous} ≡ IsContinuous(τ⇩_{1},τ⇩_{2},g)" fixes cl⇩_{1}defines cl1_def [simp]: "cl⇩_{1}(A) ≡ Closure(A,τ⇩_{1})" fixes cl⇩_{2}defines cl2_def [simp]: "cl⇩_{2}(A) ≡ Closure(A,τ⇩_{2})" text‹First we show that theorems proven in locale ‹topology0› are valid when applied to topologies $\tau_1$ and $\tau_2$.› lemma (in two_top_spaces0) topol_cntxs_valid: shows "topology0(τ⇩_{1})" and "topology0(τ⇩_{2})" using tau1_is_top tau2_is_top topology0_def by auto text‹For continuous functions the inverse image of a closed set is closed.› lemma (in two_top_spaces0) TopZF_2_1_L1: assumes A1: "f {is continuous}" and A2: "D {is closed in} τ⇩_{2}" shows "f-``(D) {is closed in} τ⇩_{1}" proof - from fmapAssum have "f-``(D) ⊆ X⇩_{1}" using func1_1_L3 by simp moreover from fmapAssum have "f-``(X⇩_{2}- D) = X⇩_{1}- f-``(D)" using Pi_iff function_vimage_Diff func1_1_L4 by auto ultimately have "X⇩_{1}- f-``(X⇩_{2}- D) = f-``(D)" by auto moreover from A1 A2 have "(X⇩_{1}- f-``(X⇩_{2}- D)) {is closed in} τ⇩_{1}" using IsClosed_def IsContinuous_def topol_cntxs_valid topology0.Top_3_L9 by simp ultimately show "f-``(D) {is closed in} τ⇩_{1}" by simp qed text‹If the inverse image of every closed set is closed, then the image of a closure is contained in the closure of the image.› lemma (in two_top_spaces0) Top_ZF_2_1_L2: assumes A1: "∀D. ((D {is closed in} τ⇩_{2}) ⟶ f-``(D) {is closed in} τ⇩_{1})" and A2: "A ⊆ X⇩_{1}" shows "f``(cl⇩_{1}(A)) ⊆ cl⇩_{2}(f``(A))" proof - from fmapAssum have "f``(A) ⊆ cl⇩_{2}(f``(A))" using func1_1_L6 topol_cntxs_valid topology0.cl_contains_set by simp with fmapAssum have "f-``(f``(A)) ⊆ f-``(cl⇩_{2}(f``(A)))" by auto moreover from fmapAssum A2 have "A ⊆ f-``(f``(A))" using func1_1_L9 by simp ultimately have "A ⊆ f-``(cl⇩_{2}(f``(A)))" by auto with fmapAssum A1 have "f``(cl⇩_{1}(A)) ⊆ f``(f-``(cl⇩_{2}(f``(A))))" using func1_1_L6 func1_1_L8 IsClosed_def topol_cntxs_valid topology0.cl_is_closed topology0.Top_3_L13 by simp moreover from fmapAssum have "f``(f-``(cl⇩_{2}(f``(A)))) ⊆ cl⇩_{2}(f``(A))" using fun_is_function function_image_vimage by simp ultimately show "f``(cl⇩_{1}(A)) ⊆ cl⇩_{2}(f``(A))" by auto qed text‹If $f\left( \overline{A}\right)\subseteq \overline{f(A)}$ (the image of the closure is contained in the closure of the image), then $\overline{f^{-1}(B)}\subseteq f^{-1}\left( \overline{B} \right)$ (the inverse image of the closure contains the closure of the inverse image).› lemma (in two_top_spaces0) Top_ZF_2_1_L3: assumes A1: "∀ A. ( A ⊆ X⇩_{1}⟶ f``(cl⇩_{1}(A)) ⊆ cl⇩_{2}(f``(A)))" shows "∀B. ( B ⊆ X⇩_{2}⟶ cl⇩_{1}(f-``(B)) ⊆ f-``(cl⇩_{2}(B)) )" proof - { fix B assume "B ⊆ X⇩_{2}" from fmapAssum A1 have "f``(cl⇩_{1}(f-``(B))) ⊆ cl⇩_{2}(f``(f-``(B)))" using func1_1_L3 by simp moreover from fmapAssum ‹B ⊆ X⇩_{2}› have "cl⇩_{2}(f``(f-``(B))) ⊆ cl⇩_{2}(B)" using fun_is_function function_image_vimage func1_1_L6 topol_cntxs_valid topology0.top_closure_mono by simp ultimately have "f-``(f``(cl⇩_{1}(f-``(B)))) ⊆ f-``(cl⇩_{2}(B))" using fmapAssum fun_is_function by auto moreover from fmapAssum ‹B ⊆ X⇩_{2}› have "cl⇩_{1}(f-``(B)) ⊆ f-``(f``(cl⇩_{1}(f-``(B))))" using func1_1_L3 func1_1_L9 IsClosed_def topol_cntxs_valid topology0.cl_is_closed by simp ultimately have "cl⇩_{1}(f-``(B)) ⊆ f-``(cl⇩_{2}(B))" by auto } then show ?thesis by simp qed text‹If $\overline{f^{-1}(B)}\subseteq f^{-1}\left( \overline{B} \right)$ (the inverse image of a closure contains the closure of the inverse image), then the function is continuous. This lemma closes a series of implications in lemmas ‹ Top_ZF_2_1_L1›, ‹ Top_ZF_2_1_L2› and ‹ Top_ZF_2_1_L3› showing equivalence of four definitions of continuity.› lemma (in two_top_spaces0) Top_ZF_2_1_L4: assumes A1: "∀B. ( B ⊆ X⇩_{2}⟶ cl⇩_{1}(f-``(B)) ⊆ f-``(cl⇩_{2}(B)) )" shows "f {is continuous}" proof - { fix U assume "U ∈ τ⇩_{2}" then have "(X⇩_{2}- U) {is closed in} τ⇩_{2}" using topol_cntxs_valid topology0.Top_3_L9 by simp moreover have "X⇩_{2}- U ⊆ ⋃τ⇩_{2}" by auto ultimately have "cl⇩_{2}(X⇩_{2}- U) = X⇩_{2}- U" using topol_cntxs_valid topology0.Top_3_L8 by simp moreover from A1 have "cl⇩_{1}(f-``(X⇩_{2}- U)) ⊆ f-``(cl⇩_{2}(X⇩_{2}- U))" by auto ultimately have "cl⇩_{1}(f-``(X⇩_{2}- U)) ⊆ f-``(X⇩_{2}- U)" by simp moreover from fmapAssum have "f-``(X⇩_{2}- U) ⊆ cl⇩_{1}(f-``(X⇩_{2}- U))" using func1_1_L3 topol_cntxs_valid topology0.cl_contains_set by simp ultimately have "f-``(X⇩_{2}- U) {is closed in} τ⇩_{1}" using fmapAssum func1_1_L3 topol_cntxs_valid topology0.Top_3_L8 by auto with fmapAssum have "f-``(U) ∈ τ⇩_{1}" using fun_is_function function_vimage_Diff func1_1_L4 func1_1_L3 IsClosed_def double_complement by simp } then have "∀U∈τ⇩_{2}. f-``(U) ∈ τ⇩_{1}" by simp then show ?thesis using IsContinuous_def by simp qed text‹For continuous functions the closure of the inverse image is contained in the inverse image of the closure. This is a shortcut through a series of implications provided by ‹TopZF_2_1_L1›, ‹Top_ZF_2_1_L2› and ‹Top_ZF_2_1_L3›. › corollary (in two_top_spaces0) im_cl_in_cl_im: assumes "f {is continuous}" and "B ⊆ X⇩_{2}" shows "cl⇩_{1}(f-``(B)) ⊆ f-``(cl⇩_{2}(B))" using assms TopZF_2_1_L1 Top_ZF_2_1_L2 Top_ZF_2_1_L3 by simp text‹Another condition for continuity: it is sufficient to check if the inverse image of every set in a base is open.› lemma (in two_top_spaces0) Top_ZF_2_1_L5: assumes A1: "B {is a base for} τ⇩_{2}" and A2: "∀U∈B. f-``(U) ∈ τ⇩_{1}" shows "f {is continuous}" proof - { fix V assume A3: "V ∈ τ⇩_{2}" with A1 obtain A where "A ⊆ B" "V = ⋃A" using IsAbaseFor_def by auto with A2 have "{f-``(U). U∈A} ⊆ τ⇩_{1}" by auto with tau1_is_top have "⋃ {f-``(U). U∈A} ∈ τ⇩_{1}" using IsATopology_def by simp moreover from ‹A ⊆ B› ‹V = ⋃A› have "f-``(V) = ⋃{f-``(U). U∈A}" by auto ultimately have "f-``(V) ∈ τ⇩_{1}" by simp } then show "f {is continuous}" using IsContinuous_def by simp qed text‹We can strenghten the previous lemma: it is sufficient to check if the inverse image of every set in a subbase is open. The proof is rather awkward, as usual when we deal with general intersections. We have to keep track of the case when the collection is empty.› lemma (in two_top_spaces0) Top_ZF_2_1_L6: assumes A1: "B {is a subbase for} τ⇩_{2}" and A2: "∀U∈B. f-``(U) ∈ τ⇩_{1}" shows "f {is continuous}" proof - let ?C = "{⋂A. A ∈ FinPow(B)}" from A1 have "?C {is a base for} τ⇩_{2}" using IsAsubBaseFor_def by simp moreover have "∀U∈?C. f-``(U) ∈ τ⇩_{1}" proof fix U assume "U∈?C" { assume "f-``(U) = 0" with tau1_is_top have "f-``(U) ∈ τ⇩_{1}" using empty_open by simp } moreover { assume "f-``(U) ≠ 0" then have "U≠0" by (rule func1_1_L13) moreover from ‹U∈?C› obtain A where "A ∈ FinPow(B)" and "U = ⋂A" by auto ultimately have "⋂A≠0" by simp then have "A≠0" by (rule inter_nempty_nempty) then have "{f-``(W). W∈A} ≠ 0" by simp moreover from A2 ‹A ∈ FinPow(B)› have "{f-``(W). W∈A} ∈ FinPow(τ⇩_{1})" by (rule fin_image_fin) ultimately have "⋂{f-``(W). W∈A} ∈ τ⇩_{1}" using topol_cntxs_valid topology0.fin_inter_open_open by simp moreover from ‹A ∈ FinPow(B)› have "A ⊆ B" using FinPow_def by simp with tau2_is_top A1 have "A ⊆ Pow(X⇩_{2})" using IsAsubBaseFor_def IsATopology_def by auto with fmapAssum ‹A≠0› ‹U = ⋂A› have "f-``(U) = ⋂{f-``(W). W∈A}" using func1_1_L12 by simp ultimately have "f-``(U) ∈ τ⇩_{1}" by simp } ultimately show "f-``(U) ∈ τ⇩_{1}" by blast qed ultimately show "f {is continuous}" using Top_ZF_2_1_L5 by simp qed text‹A dual of ‹ Top_ZF_2_1_L5›: a function that maps base sets to open sets is open.› lemma (in two_top_spaces0) base_image_open: assumes A1: "ℬ {is a base for} τ⇩_{1}" and A2: "∀B∈ℬ. f``(B) ∈ τ⇩_{2}" and A3: "U∈τ⇩_{1}" shows "f``(U) ∈ τ⇩_{2}" proof - from A1 A3 obtain ℰ where "ℰ ∈ Pow(ℬ)" and "U = ⋃ℰ" using Top_1_2_L1 by blast with A1 have "f``(U) = ⋃{f``(E). E ∈ ℰ}" using Top_1_2_L5 fmapAssum image_of_Union by auto moreover from A2 ‹ℰ ∈ Pow(ℬ)› have "{f``(E). E ∈ ℰ} ∈ Pow(τ⇩_{2})" by auto then have "⋃{f``(E). E ∈ ℰ} ∈ τ⇩_{2}" using tau2_is_top IsATopology_def by simp ultimately show ?thesis using tau2_is_top IsATopology_def by auto qed text‹A composition of two continuous functions is continuous.› lemma comp_cont: assumes "IsContinuous(T,S,f)" and "IsContinuous(S,R,g)" shows "IsContinuous(T,R,g O f)" using assms IsContinuous_def vimage_comp by simp text‹A composition of three continuous functions is continuous.› lemma comp_cont3: assumes "IsContinuous(T,S,f)" and "IsContinuous(S,R,g)" and "IsContinuous(R,P,h)" shows "IsContinuous(T,P,h O g O f)" using assms IsContinuous_def vimage_comp by simp text‹The graph of a continuous function into a Hausdorff topological space is closed in the product topology. Recall that in ZF a function is the same as its graph.› lemma (in two_top_spaces0) into_T2_graph_closed: assumes "f {is continuous}" "τ⇩_{2}{is T⇩_{2}}" shows "f {is closed in} ProductTopology(τ⇩_{1},τ⇩_{2})" proof - from fmapAssum have "f = {⟨x,f`(x)⟩. x∈X⇩_{1}}" using fun_is_set_of_pairs by simp let ?f⇩_{c}= "X⇩_{1}×X⇩_{2}- f" have "?f⇩_{c}∈ ProductTopology(τ⇩_{1},τ⇩_{2})" proof - { fix p assume "p∈?f⇩_{c}" then have "p ∈ X⇩_{1}×X⇩_{2}" and "p ∉ f" by auto from ‹p∈X⇩_{1}×X⇩_{2}› obtain x y where "x∈X⇩_{1}" "y∈X⇩_{2}" "p = ⟨x,y⟩" by auto have "y≠f`(x)" proof - { assume "y=f`(x)" with ‹x∈X⇩_{1}› ‹p = ⟨x,y⟩› have "p ∈ {⟨x,f`(x)⟩. x∈X⇩_{1}}" by auto with ‹f = {⟨x,f`(x)⟩. x∈X⇩_{1}}› ‹p ∉ f› have False by auto } thus "y≠f`(x)" by auto qed from fmapAssum ‹x∈X⇩_{1}› have "f`(x) ∈ X⇩_{2}" by (rule apply_funtype) with ‹y∈X⇩_{2}› have "y∈⋃τ⇩_{2}" "f`(x) ∈ ⋃τ⇩_{2}" by auto with assms(2) ‹y≠f`(x)› obtain U V where "U∈τ⇩_{2}" "V∈τ⇩_{2}" "y∈U" "f`(x)∈V" "U∩V = 0" unfolding isT2_def by blast let ?W = "f-``(V)" have "?W∈τ⇩_{1}" "?W⊆X⇩_{1}" "U⊆X⇩_{2}" "x∈?W" "p∈?W×U" "f``(?W) ⊆ V" proof - from assms(1) have "IsContinuous(τ⇩_{1},τ⇩_{2},f)" by simp with ‹V∈τ⇩_{2}› ‹U∈τ⇩_{2}› show "?W∈τ⇩_{1}" "?W⊆X⇩_{1}" "U⊆X⇩_{2}" unfolding IsContinuous_def by auto from fmapAssum ‹x∈X⇩_{1}› ‹f`(x)∈V› show "x∈?W" using func1_1_L15 by auto with ‹y∈U› ‹y∈U› ‹p = ⟨x,y⟩› show "p∈?W×U" by simp from fmapAssum show "f``(?W) ⊆ V" using fun_is_fun function_image_vimage by simp qed from fmapAssum ‹U∩V = 0› ‹?W⊆X⇩_{1}› ‹U⊆X⇩_{2}› have "?W×U ⊆ ?f⇩_{c}" using vimage_prod_dis_graph by blast with ‹?W∈τ⇩_{1}› ‹U∈τ⇩_{2}› ‹p∈?W×U› have "∃W∈τ⇩_{1}.∃U∈τ⇩_{2}. p∈W×U ∧ W×U ⊆ ?f⇩_{c}" by blast } with tau1_is_top tau2_is_top show "?f⇩_{c}∈ ProductTopology(τ⇩_{1},τ⇩_{2})" using point_neighb_prod_top by simp qed with fmapAssum tau1_is_top tau2_is_top show ?thesis using fun_subset_prod Top_1_4_T1(3) unfolding IsClosed_def by auto qed subsection‹Homeomorphisms› text‹This section studies ''homeomorphisms'' - continous bijections whose inverses are also continuous. Notions that are preserved by (commute with) homeomorphisms are called ''topological invariants''.› text‹Homeomorphism is a bijection that preserves open sets.› definition "IsAhomeomorphism(T,S,f) ≡ f ∈ bij(⋃T,⋃S) ∧ IsContinuous(T,S,f) ∧ IsContinuous(S,T,converse(f))" text‹Inverse (converse) of a homeomorphism is a homeomorphism.› lemma homeo_inv: assumes "IsAhomeomorphism(T,S,f)" shows "IsAhomeomorphism(S,T,converse(f))" using assms IsAhomeomorphism_def bij_converse_bij bij_converse_converse by auto text‹Homeomorphisms are open maps.› lemma homeo_open: assumes "IsAhomeomorphism(T,S,f)" and "U∈T" shows "f``(U) ∈ S" using assms image_converse IsAhomeomorphism_def IsContinuous_def by simp text‹A continuous bijection that is an open map is a homeomorphism.› lemma bij_cont_open_homeo: assumes "f ∈ bij(⋃T,⋃S)" and "IsContinuous(T,S,f)" and "∀U∈T. f``(U) ∈ S" shows "IsAhomeomorphism(T,S,f)" using assms image_converse IsAhomeomorphism_def IsContinuous_def by auto text‹A continuous bijection that maps base to open sets is a homeomorphism.› lemma (in two_top_spaces0) bij_base_open_homeo: assumes A1: "f ∈ bij(X⇩_{1},X⇩_{2})" and A2: "ℬ {is a base for} τ⇩_{1}" and A3: "𝒞 {is a base for} τ⇩_{2}" and A4: "∀U∈𝒞. f-``(U) ∈ τ⇩_{1}" and A5: "∀V∈ℬ. f``(V) ∈ τ⇩_{2}" shows "IsAhomeomorphism(τ⇩_{1},τ⇩_{2},f)" using assms tau2_is_top tau1_is_top bij_converse_bij bij_is_fun two_top_spaces0_def image_converse two_top_spaces0.Top_ZF_2_1_L5 IsAhomeomorphism_def by simp text‹A bijection that maps base to base is a homeomorphism.› lemma (in two_top_spaces0) bij_base_homeo: assumes A1: "f ∈ bij(X⇩_{1},X⇩_{2})" and A2: "ℬ {is a base for} τ⇩_{1}" and A3: "{f``(B). B∈ℬ} {is a base for} τ⇩_{2}" shows "IsAhomeomorphism(τ⇩_{1},τ⇩_{2},f)" proof - note A1 moreover have "f {is continuous}" proof - { fix C assume "C ∈ {f``(B). B∈ℬ}" then obtain B where "B∈ℬ" and I: "C = f``(B)" by auto with A2 have "B ⊆ X⇩_{1}" using Top_1_2_L5 by auto with A1 A2 ‹B∈ℬ› I have "f-``(C) ∈ τ⇩_{1}" using bij_def inj_vimage_image base_sets_open by auto } hence "∀C ∈ {f``(B). B∈ℬ}. f-``(C) ∈ τ⇩_{1}" by auto with A3 show ?thesis by (rule Top_ZF_2_1_L5) qed moreover from A3 have "∀B∈ℬ. f``(B) ∈ τ⇩_{2}" using base_sets_open by auto with A2 have "∀U∈τ⇩_{1}. f``(U) ∈ τ⇩_{2}" using base_image_open by simp ultimately show ?thesis using bij_cont_open_homeo by simp qed text‹Interior is a topological invariant.› theorem int_top_invariant: assumes A1: "A⊆⋃T" and A2: "IsAhomeomorphism(T,S,f)" shows "f``(Interior(A,T)) = Interior(f``(A),S)" proof - let ?𝒜 = "{U∈T. U⊆A}" have I: "{f``(U). U∈?𝒜} = {V∈S. V ⊆ f``(A)}" proof from A2 show "{f``(U). U∈?𝒜} ⊆ {V∈S. V ⊆ f``(A)}" using homeo_open by auto { fix V assume "V ∈ {V∈S. V ⊆ f``(A)}" hence "V∈S" and II: "V ⊆ f``(A)" by auto let ?U = "f-``(V)" from II have "?U ⊆ f-``(f``(A))" by auto moreover from assms have "f-``(f``(A)) = A" using IsAhomeomorphism_def bij_def inj_vimage_image by auto moreover from A2 ‹V∈S› have "?U∈T" using IsAhomeomorphism_def IsContinuous_def by simp moreover from ‹V∈S› have "V ⊆ ⋃S" by auto with A2 have "V = f``(?U)" using IsAhomeomorphism_def bij_def surj_image_vimage by auto ultimately have "V ∈ {f``(U). U∈?𝒜}" by auto } thus "{V∈S. V ⊆ f``(A)} ⊆ {f``(U). U∈?𝒜}" by auto qed have "f``(Interior(A,T)) = f``(⋃?𝒜)" unfolding Interior_def by simp also from A2 have "… = ⋃{f``(U). U∈?𝒜}" using IsAhomeomorphism_def bij_def inj_def image_of_Union by auto also from I have "… = Interior(f``(A),S)" unfolding Interior_def by simp finally show ?thesis by simp qed subsection‹Topologies induced by mappings› text‹In this section we consider various ways a topology may be defined on a set that is the range (or the domain) of a function whose domain (or range) is a topological space. › text‹A bijection from a topological space induces a topology on the range.› theorem bij_induced_top: assumes A1: "T {is a topology}" and A2: "f ∈ bij(⋃T,Y)" shows "{f``(U). U∈T} {is a topology}" and "{ {f`(x).x∈U}. U∈T} {is a topology}" and "(⋃{f``(U). U∈T}) = Y" and "IsAhomeomorphism(T, {f``(U). U∈T},f)" proof - from A2 have "f ∈ inj(⋃T,Y)" using bij_def by simp then have "f:⋃T→Y" using inj_def by simp let ?S = "{f``(U). U∈T}" { fix M assume "M ∈ Pow(?S)" let ?M⇩_{T}= "{f-``(V). V∈M}" have "?M⇩_{T}⊆ T" proof fix W assume "W∈?M⇩_{T}" then obtain V where "V∈M" and I: "W = f-``(V)" by auto with ‹M ∈ Pow(?S)› have "V∈?S" by auto then obtain U where "U∈T" and "V = f``(U)" by auto with I have "W = f-``(f``(U))" by simp with ‹f ∈ inj(⋃T,Y)› ‹U∈T› have "W = U" using inj_vimage_image by blast with ‹U∈T› show "W∈T" by simp qed with A1 have "(⋃?M⇩_{T}) ∈ T" using IsATopology_def by simp hence "f``(⋃?M⇩_{T}) ∈ ?S" by auto moreover have "f``(⋃?M⇩_{T}) = ⋃M" proof - from ‹f:⋃T→Y› ‹?M⇩_{T}⊆ T› have "f``(⋃?M⇩_{T}) = ⋃{f``(U). U∈?M⇩_{T}}" using image_of_Union by auto moreover have "{f``(U). U∈?M⇩_{T}} = M" proof - from ‹f:⋃T→Y› have "∀U∈T. f``(U) ⊆ Y" using func1_1_L6 by simp with ‹M ∈ Pow(?S)› have "M ⊆ Pow(Y)" by auto with A2 show "{f``(U). U∈?M⇩_{T}} = M" using bij_def surj_subsets by auto qed ultimately show "f``(⋃?M⇩_{T}) = ⋃M" by simp qed ultimately have "⋃M ∈ ?S" by auto } then have "∀M∈Pow(?S). ⋃M ∈ ?S" by auto moreover { fix U V assume "U∈?S" "V∈?S" then obtain U⇩_{T}V⇩_{T}where "U⇩_{T}∈ T" "V⇩_{T}∈ T" and I: "U = f``(U⇩_{T})" "V = f``(V⇩_{T})" by auto with A1 have "U⇩_{T}∩V⇩_{T}∈ T" using IsATopology_def by simp hence "f``(U⇩_{T}∩V⇩_{T}) ∈ ?S" by auto moreover have "f``(U⇩_{T}∩V⇩_{T}) = U∩V" proof - from ‹U⇩_{T}∈ T› ‹V⇩_{T}∈ T› have "U⇩_{T}⊆ ⋃T" "V⇩_{T}⊆ ⋃T" using bij_def by auto with ‹f ∈ inj(⋃T,Y)› I show "f``(U⇩_{T}∩V⇩_{T}) = U∩V" using inj_image_inter by simp qed ultimately have "U∩V ∈ ?S" by simp } then have "∀U∈?S. ∀V∈?S. U∩V ∈ ?S" by auto ultimately show "?S {is a topology}" using IsATopology_def by simp moreover from ‹f:⋃T→Y› have "∀U∈T. f``(U) = {f`(x).x∈U}" using func_imagedef by blast ultimately show "{ {f`(x).x∈U}. U∈T} {is a topology}" by simp show "⋃?S = Y" proof from ‹f:⋃T→Y› have "∀U∈T. f``(U) ⊆ Y" using func1_1_L6 by simp thus "⋃?S ⊆ Y" by auto from A1 have "f``(⋃T) ⊆ ⋃?S" using IsATopology_def by auto with A2 show "Y ⊆ ⋃?S" using bij_def surj_range_image_domain by auto qed show "IsAhomeomorphism(T,?S,f)" proof - from A2 ‹⋃?S = Y› have "f ∈ bij(⋃T,⋃?S)" by simp moreover have "IsContinuous(T,?S,f)" proof - { fix V assume "V∈?S" then obtain U where "U∈T" and "V = f``(U)" by auto hence "U ⊆ ⋃T" and "f-``(V) = f-``(f``(U))" by auto with ‹f ∈ inj(⋃T,Y)› ‹U∈T› have "f-``(V) ∈ T" using inj_vimage_image by simp } then show "IsContinuous(T,?S,f)" unfolding IsContinuous_def by auto qed ultimately show"IsAhomeomorphism(T,?S,f)" using bij_cont_open_homeo by auto qed qed subsection‹Partial functions and continuity› text‹Suppose we have two topologies $\tau_1,\tau_2$ on sets $X_i=\bigcup\tau_i, i=1,2$. Consider some function $f:A\rightarrow X_2$, where $A\subseteq X_1$ (we will call such function ''partial''). In such situation we have two natural possibilities for the pairs of topologies with respect to which this function may be continuous. One is obviously the original $\tau_1,\tau_2$ and in the second one the first element of the pair is the topology relative to the domain of the function: $\{A\cap U | U \in \tau_1\}$. These two possibilities are not exactly the same and the goal of this section is to explore the differences.› text‹If a function is continuous, then its restriction is continous in relative topology.› lemma (in two_top_spaces0) restr_cont: assumes A1: "A ⊆ X⇩_{1}" and A2: "f {is continuous}" shows "IsContinuous(τ⇩_{1}{restricted to} A, τ⇩_{2},restrict(f,A))" proof - let ?g = "restrict(f,A)" { fix U assume "U ∈ τ⇩_{2}" with A2 have "f-``(U) ∈ τ⇩_{1}" using IsContinuous_def by simp moreover from A1 have "?g-``(U) = f-``(U) ∩ A" using fmapAssum func1_2_L1 by simp ultimately have "?g-``(U) ∈ (τ⇩_{1}{restricted to} A)" using RestrictedTo_def by auto } then show ?thesis using IsContinuous_def by simp qed text‹If a function is continuous, then it is continuous when we restrict the topology on the range to the image of the domain.› lemma (in two_top_spaces0) restr_image_cont: assumes A1: "f {is continuous}" shows "IsContinuous(τ⇩_{1}, τ⇩_{2}{restricted to} f``(X⇩_{1}),f)" proof - have "∀U ∈ τ⇩_{2}{restricted to} f``(X⇩_{1}). f-``(U) ∈ τ⇩_{1}" proof fix U assume "U ∈ τ⇩_{2}{restricted to} f``(X⇩_{1})" then obtain V where "V ∈ τ⇩_{2}" and "U = V ∩ f``(X⇩_{1})" using RestrictedTo_def by auto with A1 show "f-``(U) ∈ τ⇩_{1}" using fmapAssum inv_im_inter_im IsContinuous_def by simp qed then show ?thesis using IsContinuous_def by simp qed text‹A combination of ‹restr_cont› and ‹restr_image_cont›.› lemma (in two_top_spaces0) restr_restr_image_cont: assumes A1: "A ⊆ X⇩_{1}" and A2: "f {is continuous}" and A3: "g = restrict(f,A)" and A4: "τ⇩_{3}= τ⇩_{1}{restricted to} A" shows "IsContinuous(τ⇩_{3}, τ⇩_{2}{restricted to} g``(A),g)" proof - from A1 A4 have "⋃τ⇩_{3}= A" using union_restrict by auto have "two_top_spaces0(τ⇩_{3}, τ⇩_{2}, g)" proof - from A4 have "τ⇩_{3}{is a topology}" and "τ⇩_{2}{is a topology}" using tau1_is_top tau2_is_top topology0_def topology0.Top_1_L4 by auto moreover from A1 A3 ‹⋃τ⇩_{3}= A› have "g: ⋃τ⇩_{3}→ ⋃τ⇩_{2}" using fmapAssum restrict_type2 by simp ultimately show ?thesis using two_top_spaces0_def by simp qed moreover from assms have "IsContinuous(τ⇩_{3}, τ⇩_{2}, g)" using restr_cont by simp ultimately have "IsContinuous(τ⇩_{3}, τ⇩_{2}{restricted to} g``(⋃τ⇩_{3}),g)" by (rule two_top_spaces0.restr_image_cont) moreover note ‹⋃τ⇩_{3}= A› ultimately show ?thesis by simp qed text‹We need a context similar to ‹two_top_spaces0› but without the global function $f:X_1\rightarrow X_2$.› locale two_top_spaces1 = fixes τ⇩_{1}assumes tau1_is_top: "τ⇩_{1}{is a topology}" fixes τ⇩_{2}assumes tau2_is_top: "τ⇩_{2}{is a topology}" fixes X⇩_{1}defines X1_def [simp]: "X⇩_{1}≡ ⋃τ⇩_{1}" fixes X⇩_{2}defines X2_def [simp]: "X⇩_{2}≡ ⋃τ⇩_{2}" text‹If a partial function $g:X_1\supseteq A\rightarrow X_2$ is continuous with respect to $(\tau_1,\tau_2)$, then $A$ is open (in $\tau_1$) and the function is continuous in the relative topology.› lemma (in two_top_spaces1) partial_fun_cont: assumes A1: "g:A→X⇩_{2}" and A2: "IsContinuous(τ⇩_{1},τ⇩_{2},g)" shows "A ∈ τ⇩_{1}" and "IsContinuous(τ⇩_{1}{restricted to} A, τ⇩_{2}, g)" proof - from A2 have "g-``(X⇩_{2}) ∈ τ⇩_{1}" using tau2_is_top IsATopology_def IsContinuous_def by simp with A1 show "A ∈ τ⇩_{1}" using func1_1_L4 by simp { fix V assume "V ∈ τ⇩_{2}" with A2 have "g-``(V) ∈ τ⇩_{1}" using IsContinuous_def by simp moreover from A1 have "g-``(V) ⊆ A" using func1_1_L3 by simp hence "g-``(V) = A ∩ g-``(V)" by auto ultimately have "g-``(V) ∈ (τ⇩_{1}{restricted to} A)" using RestrictedTo_def by auto } then show "IsContinuous(τ⇩_{1}{restricted to} A, τ⇩_{2}, g)" using IsContinuous_def by simp qed text‹For partial function defined on open sets continuity in the whole and relative topologies are the same.› lemma (in two_top_spaces1) part_fun_on_open_cont: assumes A1: "g:A→X⇩_{2}" and A2: "A ∈ τ⇩_{1}" shows "IsContinuous(τ⇩_{1},τ⇩_{2},g) ⟷ IsContinuous(τ⇩_{1}{restricted to} A, τ⇩_{2}, g)" proof assume "IsContinuous(τ⇩_{1},τ⇩_{2},g)" with A1 show "IsContinuous(τ⇩_{1}{restricted to} A, τ⇩_{2}, g)" using partial_fun_cont by simp next assume I: "IsContinuous(τ⇩_{1}{restricted to} A, τ⇩_{2}, g)" { fix V assume "V ∈ τ⇩_{2}" with I have "g-``(V) ∈ (τ⇩_{1}{restricted to} A)" using IsContinuous_def by simp then obtain W where "W ∈ τ⇩_{1}" and "g-``(V) = A∩W" using RestrictedTo_def by auto with A2 have "g-``(V) ∈ τ⇩_{1}" using tau1_is_top IsATopology_def by simp } then show "IsContinuous(τ⇩_{1},τ⇩_{2},g)" using IsContinuous_def by simp qed subsection‹Product topology and continuity› text‹We start with three topological spaces $(\tau_1,X_1), (\tau_2,X_2)$ and $(\tau_3,X_3)$ and a function $f:X_1\times X_2\rightarrow X_3$. We will study the properties of $f$ with respect to the product topology $\tau_1\times \tau_2$ and $\tau_3$. This situation is similar as in locale ‹two_top_spaces0› but the first topological space is assumed to be a product of two topological spaces. › text‹First we define a locale with three topological spaces.› locale prod_top_spaces0 = fixes τ⇩_{1}assumes tau1_is_top: "τ⇩_{1}{is a topology}" fixes τ⇩_{2}assumes tau2_is_top: "τ⇩_{2}{is a topology}" fixes τ⇩_{3}assumes tau3_is_top: "τ⇩_{3}{is a topology}" fixes X⇩_{1}defines X1_def [simp]: "X⇩_{1}≡ ⋃τ⇩_{1}" fixes X⇩_{2}defines X2_def [simp]: "X⇩_{2}≡ ⋃τ⇩_{2}" fixes X⇩_{3}defines X3_def [simp]: "X⇩_{3}≡ ⋃τ⇩_{3}" fixes η defines eta_def [simp]: "η ≡ ProductTopology(τ⇩_{1},τ⇩_{2})" text‹Fixing the first variable in a two-variable continuous function results in a continuous function.› lemma (in prod_top_spaces0) fix_1st_var_cont: assumes "f: X⇩_{1}×X⇩_{2}→X⇩_{3}" and "IsContinuous(η,τ⇩_{3},f)" and "x∈X⇩_{1}" shows "IsContinuous(τ⇩_{2},τ⇩_{3},Fix1stVar(f,x))" using assms fix_1st_var_vimage IsContinuous_def tau1_is_top tau2_is_top prod_sec_open1 by simp text‹Fixing the second variable in a two-variable continuous function results in a continuous function.› lemma (in prod_top_spaces0) fix_2nd_var_cont: assumes "f: X⇩_{1}×X⇩_{2}→X⇩_{3}" and "IsContinuous(η,τ⇩_{3},f)" and "y∈X⇩_{2}" shows "IsContinuous(τ⇩_{1},τ⇩_{3},Fix2ndVar(f,y))" using assms fix_2nd_var_vimage IsContinuous_def tau1_is_top tau2_is_top prod_sec_open2 by simp text‹Having two constinuous mappings we can construct a third one on the cartesian product of the domains.› lemma cart_prod_cont: assumes A1: "τ⇩_{1}{is a topology}" "τ⇩_{2}{is a topology}" and A2: "η⇩_{1}{is a topology}" "η⇩_{2}{is a topology}" and A3a: "f⇩_{1}:⋃τ⇩_{1}→⋃η⇩_{1}" and A3b: "f⇩_{2}:⋃τ⇩_{2}→⋃η⇩_{2}" and A4: "IsContinuous(τ⇩_{1},η⇩_{1},f⇩_{1})" "IsContinuous(τ⇩_{2},η⇩_{2},f⇩_{2})" and A5: "g = {⟨p,⟨f⇩_{1}`(fst(p)),f⇩_{2}`(snd(p))⟩⟩. p ∈ ⋃τ⇩_{1}×⋃τ⇩_{2}}" shows "IsContinuous(ProductTopology(τ⇩_{1},τ⇩_{2}),ProductTopology(η⇩_{1},η⇩_{2}),g)" proof - let ?τ = "ProductTopology(τ⇩_{1},τ⇩_{2})" let ?η = "ProductTopology(η⇩_{1},η⇩_{2})" let ?X⇩_{1}= "⋃τ⇩_{1}" let ?X⇩_{2}= "⋃τ⇩_{2}" let ?Y⇩_{1}= "⋃η⇩_{1}" let ?Y⇩_{2}= "⋃η⇩_{2}" let ?B = "ProductCollection(η⇩_{1},η⇩_{2})" from A1 A2 have "?τ {is a topology}" and "?η {is a topology}" using Top_1_4_T1 by auto moreover have "g: ?X⇩_{1}×?X⇩_{2}→ ?Y⇩_{1}×?Y⇩_{2}" proof - { fix p assume "p ∈ ?X⇩_{1}×?X⇩_{2}" hence "fst(p) ∈ ?X⇩_{1}" and "snd(p) ∈ ?X⇩_{2}" by auto from A3a ‹fst(p) ∈ ?X⇩_{1}› have "f⇩_{1}`(fst(p)) ∈ ?Y⇩_{1}" by (rule apply_funtype) moreover from A3b ‹snd(p) ∈ ?X⇩_{2}› have "f⇩_{2}`(snd(p)) ∈ ?Y⇩_{2}" by (rule apply_funtype) ultimately have "⟨f⇩_{1}`(fst(p)),f⇩_{2}`(snd(p))⟩ ∈ ⋃η⇩_{1}×⋃η⇩_{2}" by auto } hence "∀p ∈ ?X⇩_{1}×?X⇩_{2}. ⟨f⇩_{1}`(fst(p)),f⇩_{2}`(snd(p))⟩ ∈ ?Y⇩_{1}×?Y⇩_{2}" by simp with A5 show "g: ?X⇩_{1}×?X⇩_{2}→ ?Y⇩_{1}×?Y⇩_{2}" using ZF_fun_from_total by simp qed moreover from A1 A2 have "⋃?τ = ?X⇩_{1}×?X⇩_{2}" and "⋃?η = ?Y⇩_{1}×?Y⇩_{2}" using Top_1_4_T1 by auto ultimately have "two_top_spaces0(?τ,?η,g)" using two_top_spaces0_def by simp moreover from A2 have "?B {is a base for} ?η" using Top_1_4_T1 by simp moreover have "∀U∈?B. g-``(U) ∈ ?τ" proof fix U assume "U∈?B" then obtain V W where "V ∈ η⇩_{1}" "W ∈ η⇩_{2}" and "U = V×W" using ProductCollection_def by auto with A3a A3b A5 have "g-``(U) = f⇩_{1}-``(V) × f⇩_{2}-``(W)" using cart_prod_fun_vimage by simp moreover from A1 A4 ‹V ∈ η⇩_{1}› ‹W ∈ η⇩_{2}› have "f⇩_{1}-``(V) × f⇩_{2}-``(W) ∈ ?τ" using IsContinuous_def prod_open_open_prod by simp ultimately show "g-``(U) ∈ ?τ" by simp qed ultimately show ?thesis using two_top_spaces0.Top_ZF_2_1_L5 by simp qed text‹A reformulation of the ‹cart_prod_cont› lemma above in slightly different notation.› theorem (in two_top_spaces0) product_cont_functions: assumes "f:X⇩_{1}→X⇩_{2}" "g:⋃τ⇩_{3}→⋃τ⇩_{4}" "IsContinuous(τ⇩_{1},τ⇩_{2},f)" "IsContinuous(τ⇩_{3},τ⇩_{4},g)" "τ⇩_{4}{is a topology}" "τ⇩_{3}{is a topology}" shows "IsContinuous(ProductTopology(τ⇩_{1},τ⇩_{3}),ProductTopology(τ⇩_{2},τ⇩_{4}),{⟨⟨x,y⟩,⟨f`x,g`y⟩⟩. ⟨x,y⟩∈X⇩_{1}×⋃τ⇩_{3}})" proof - have "{⟨⟨x,y⟩,⟨f`x,g`y⟩⟩. ⟨x,y⟩∈X⇩_{1}×⋃τ⇩_{3}} = {⟨p,⟨f`(fst(p)),g`(snd(p))⟩⟩. p ∈ X⇩_{1}×⋃τ⇩_{3}}" by force with tau1_is_top tau2_is_top assms show ?thesis using cart_prod_cont by simp qed text‹A special case of ‹cart_prod_cont› when the function acting on the second axis is the identity.› lemma cart_prod_cont1: assumes A1: "τ⇩_{1}{is a topology}" and A1a: "τ⇩_{2}{is a topology}" and A2: "η⇩_{1}{is a topology}" and A3: "f⇩_{1}:⋃τ⇩_{1}→⋃η⇩_{1}" and A4: "IsContinuous(τ⇩_{1},η⇩_{1},f⇩_{1})" and A5: "g = {⟨p, ⟨f⇩_{1}`(fst(p)),snd(p)⟩⟩. p ∈ ⋃τ⇩_{1}×⋃τ⇩_{2}}" shows "IsContinuous(ProductTopology(τ⇩_{1},τ⇩_{2}),ProductTopology(η⇩_{1},τ⇩_{2}),g)" proof - let ?f⇩_{2}= "id(⋃τ⇩_{2})" have "∀x∈⋃τ⇩_{2}. ?f⇩_{2}`(x) = x" using id_conv by blast hence I: "∀p ∈ ⋃τ⇩_{1}×⋃τ⇩_{2}. snd(p) = ?f⇩_{2}`(snd(p))" by simp note A1 A1a A2 A1a A3 moreover have "?f⇩_{2}:⋃τ⇩_{2}→⋃τ⇩_{2}" using id_type by simp moreover note A4 moreover have "IsContinuous(τ⇩_{2},τ⇩_{2},?f⇩_{2})" using id_cont by simp moreover have "g = {⟨p, ⟨f⇩_{1}`(fst(p)),?f⇩_{2}`(snd(p))⟩ ⟩. p ∈ ⋃τ⇩_{1}×⋃τ⇩_{2}}" proof from A5 I show "g ⊆ {⟨p, ⟨f⇩_{1}`(fst(p)),?f⇩_{2}`(snd(p))⟩⟩. p ∈ ⋃τ⇩_{1}×⋃τ⇩_{2}}" by auto from A5 I show "{⟨p, ⟨f⇩_{1}`(fst(p)),?f⇩_{2}`(snd(p))⟩⟩. p ∈ ⋃τ⇩_{1}×⋃τ⇩_{2}} ⊆ g" by auto qed ultimately show ?thesis by (rule cart_prod_cont) qed text‹Having two continuous mappings $f,g$ we can construct a third one with values in the cartesian product of the codomains of $f,g$, defined by $x\mapsto \langle f(x),g(x) \rangle$. › lemma (in prod_top_spaces0) cont_funcs_prod: assumes "f:X⇩_{1}→X⇩_{2}" "g:X⇩_{1}→X⇩_{3}" "IsContinuous(τ⇩_{1},τ⇩_{2},f)" "IsContinuous(τ⇩_{1},τ⇩_{3},g)" defines "h ≡ {⟨x,⟨f`(x),g`(x)⟩⟩. x∈X⇩_{1}}" shows "IsContinuous(τ⇩_{1},ProductTopology(τ⇩_{2},τ⇩_{3}),h)" proof - let ?B = "ProductCollection(τ⇩_{2},τ⇩_{3})" have "two_top_spaces0(τ⇩_{1},ProductTopology(τ⇩_{2},τ⇩_{3}),h)" "?B {is a base for} ProductTopology(τ⇩_{2},τ⇩_{3})" "∀W∈?B. h-``(W) ∈ τ⇩_{1}" proof - from tau1_is_top tau2_is_top tau3_is_top assms(1,2,5) show "two_top_spaces0(τ⇩_{1},ProductTopology(τ⇩_{2},τ⇩_{3}),h)" using vimage_prod Top_1_4_T1(1,3) unfolding two_top_spaces0_def by simp from tau2_is_top tau3_is_top show"?B {is a base for} ProductTopology(τ⇩_{2},τ⇩_{3})" using Top_1_4_T1(2) by simp from tau1_is_top assms show "∀W∈?B. h-``(W) ∈ τ⇩_{1}" unfolding ProductCollection_def IsContinuous_def IsATopology_def using vimage_prod by simp qed then show ?thesis by (rule two_top_spaces0.Top_ZF_2_1_L5) qed text‹Having two continuous mappings $f,g$ we can construct a third one with values in the cartesian product of the codomains of $f,g$, defined by $x\mapsto \langle f(x),g(x) \rangle$. This is essentially the same as ‹cont_funcs_prod› but formulated in a way that is sometimes easier to apply. Recall that $\tau_2 \times_t \tau_3$ is a notation for the product topology of $\tau_1$ and $\tau_2$. › lemma cont_funcs_prod1: assumes "τ⇩_{1}{is a topology}" "τ⇩_{2}{is a topology}" "τ⇩_{3}{is a topology}" and "{⟨x,p(x)⟩. x∈⋃τ⇩_{1}} ∈ Cont(τ⇩_{1},τ⇩_{2})" "{⟨x,q(x)⟩. x∈⋃τ⇩_{1}} ∈ Cont(τ⇩_{1},τ⇩_{3})" shows "{⟨x,⟨p(x),q(x)⟩⟩. x∈⋃τ⇩_{1}} ∈ Cont(τ⇩_{1},τ⇩_{2}×⇩_{t}τ⇩_{3})" proof - let ?X = "⋃τ⇩_{1}" let ?Y = "⋃τ⇩_{2}" let ?Z = "⋃τ⇩_{3}" let ?f = "{⟨x,p(x)⟩. x∈?X}" let ?g = "{⟨x,q(x)⟩. x∈?X}" let ?h = "{⟨x,⟨p(x),q(x)⟩⟩. x∈?X}" from assms(4,5) have "?f:?X→?Y" and "?g:?X→?Z" unfolding Cont_def by auto with assms(2,3) have hFun: "?h:?X→⋃(τ⇩_{2}×⇩_{t}τ⇩_{3})" using prod_fun_val(1) using Top_1_4_T1(3) by simp moreover have "IsContinuous(τ⇩_{1},τ⇩_{2}×⇩_{t}τ⇩_{3},?h)" proof - let ?B = "ProductCollection(τ⇩_{2},τ⇩_{3})" from assms(1,2,3) hFun have "two_top_spaces0(τ⇩_{1},τ⇩_{2}×⇩_{t}τ⇩_{3},?h)" using Top_1_4_T1(1) unfolding two_top_spaces0_def by simp moreover from assms(2,3) have "?B {is a base for} (τ⇩_{2}×⇩_{t}τ⇩_{3})" using Top_1_4_T1(2) by simp moreover have "∀W∈?B. ?h-``(W) ∈ τ⇩_{1}" proof - { fix W assume "W∈?B" then obtain U V where "U∈τ⇩_{2}" "V∈τ⇩_{3}" "W=U×V" unfolding ProductCollection_def by auto have "∀x∈?X. ⟨p(x),q(x)⟩ = ⟨?f`(x),?g`(x)⟩" proof - { fix x assume "x∈?X" then have "⟨p(x),q(x)⟩ = ⟨?f`(x),?g`(x)⟩" using ZF_fun_from_tot_val1 by simp } thus ?thesis by auto qed then have "?h = {⟨x,⟨?f`(x),?g`(x)⟩⟩. x∈?X}" by (rule set_comp_eq) with assms(1,4,5) ‹?f:?X→?Y› ‹?g:?X→?Z› ‹U∈τ⇩_{2}› ‹V∈τ⇩_{3}› ‹W=U×V› have "?h-``(W) ∈ τ⇩_{1}" using vimage_prod(3) unfolding Cont_def IsContinuous_def IsATopology_def by auto } thus ?thesis by simp qed ultimately show ?thesis by (rule two_top_spaces0.Top_ZF_2_1_L5) qed ultimately show ?thesis unfolding Cont_def by simp qed text‹Two continuous functions into a Hausdorff space are equal on a closed set. Note that in the lemma below $f$ is assumed to map $X_1$ to $X_2$ in the locale, we only need to add a similar assumption for the second function. › lemma (in two_top_spaces0) two_fun_eq_closed: assumes "g:X⇩_{1}→X⇩_{2}" "f {is continuous}" "g {is continuous}" "τ⇩_{2}{is T⇩_{2}}" shows "{x∈X⇩_{1}. f`(x)=g`(x)} {is closed in} τ⇩_{1}" proof - let ?D = "{x∈X⇩_{1}. f`(x)=g`(x)}" let ?h = "{⟨x,⟨f`(x),g`(x)⟩⟩. x∈X⇩_{1}}" have "?h-``({⟨y,y⟩. y∈X⇩_{2}}) {is closed in} τ⇩_{1}" proof - have "two_top_spaces0(τ⇩_{1},ProductTopology(τ⇩_{2},τ⇩_{2}),?h)" proof from tau1_is_top tau2_is_top show "τ⇩_{1}{is a topology}" and "ProductTopology(τ⇩_{2},τ⇩_{2}) {is a topology}" using Top_1_4_T1(1) by auto from tau1_is_top tau2_is_top fmapAssum assms(1) show "?h : ⋃τ⇩_{1}→ ⋃ProductTopology(τ⇩_{2}, τ⇩_{2})" using vimage_prod(1) Top_1_4_T1(3) by simp qed moreover have "IsContinuous(τ⇩_{1},ProductTopology(τ⇩_{2},τ⇩_{2}),?h)" proof - from tau1_is_top tau2_is_top have "prod_top_spaces0(τ⇩_{1},τ⇩_{2},τ⇩_{2})" unfolding prod_top_spaces0_def by simp with fmapAssum assms(1,2,3) show ?thesis using prod_top_spaces0.cont_funcs_prod by simp qed moreover from tau2_is_top assms(4) have "{⟨y,y⟩. y∈X⇩_{2}} {is closed in} ProductTopology(τ⇩_{2},τ⇩_{2})" using t2_iff_diag_closed by simp ultimately show ?thesis using two_top_spaces0.TopZF_2_1_L1 by simp qed with fmapAssum assms(1) show ?thesis using vimage_prod(4) by simp qed text‹Closure of an image of a singleton by a relation in $X\times Y$ is contained in the image of this singleton by the closure of the relation (in the product topology). Compare the proof of Metamath's theorem with the same name.› lemma imasncls: assumes "T {is a topology}" "S {is a topology}" "R ⊆ (⋃T)×(⋃S)" "x∈⋃T" shows "Closure(R``{x},S) ⊆ Closure(R,T×⇩_{t}S)``{x}" proof - let ?X = "⋃T" let ?Y = "⋃S" let ?f = "{⟨y,⟨x,y⟩⟩. y∈?Y}" from assms(3) have "R``{x} = ?f-``(R)" by blast hence "Closure(R``{x},S) = Closure(?f-``(R),S)" by simp also have "Closure(?f-``(R),S) ⊆ ?f-``(Closure(R,T×⇩_{t}S))" proof - from assms(1,2,4) have "?f ∈ Cont(S,T×⇩_{t}S)" using const_cont_sp id_cont_sp cont_funcs_prod1 by simp with assms(1,2,3) have "two_top_spaces0(S,T×⇩_{t}S,?f)" "IsContinuous(S,T×⇩_{t}S,?f)" "R ⊆ ⋃(T×⇩_{t}S)" unfolding Cont_def two_top_spaces0_def using Top_1_4_T1(1,3) by auto then show "Closure(?f-``(R),S) ⊆ ?f-``(Closure(R,T×⇩_{t}S))" using two_top_spaces0.im_cl_in_cl_im by simp qed also have "Closure(R,T×⇩_{t}S) ⊆ ?X×?Y" proof - from assms(1,2,3) have "topology0(T×⇩_{t}S)" "R ⊆ ⋃(T×⇩_{t}S)" unfolding topology0_def using Top_1_4_T1(1,3) by auto then have "Closure(R,T×⇩_{t}S) ⊆ ⋃(T×⇩_{t}S)" by (rule topology0.Top_3_L11) with assms(1,2) show ?thesis using Top_1_4_T1(3) by simp qed with assms(4) have "?f-``(Closure(R,T×⇩_{t}S)) = Closure(R,T×⇩_{t}S)``{x}" by blast finally show "Closure(R``{x},S) ⊆ Closure(R,T×⇩_{t}S)``{x}" by simp qed subsection‹Pasting lemma› text‹The classical pasting lemma states that if $U_1,U_2$ are both open (or closed) and a function is continuous when restricted to both $U_1$ and $U_2$ then it is continuous when restricted to $U_1 \cup U_2$. In this section we prove a generalization statement stating that the set $\{ U \in \tau_1 | f|_U$ is continuous $\}$ is a topology.› text‹A typical statement of the pasting lemma uses the notion of a function restricted to a set being continuous without specifying the topologies with respect to which this continuity holds. In ‹two_top_spaces0› context the notation ‹g {is continuous}› means continuity wth respect to topologies $\tau_1, \tau_2$. The next lemma is a special case of ‹partial_fun_cont› and states that if for some set $A\subseteq X_1=\bigcup \tau_1$ the function $f|_A$ is continuous (with respect to $(\tau_1, \tau_2)$), then $A$ has to be open. This clears up terminology and indicates why we need to pay attention to the issue of which topologies we talk about when we say that the restricted (to some closed set for example) function is continuos. › lemma (in two_top_spaces0) restriction_continuous1: assumes A1: "A ⊆ X⇩_{1}" and A2: "restrict(f,A) {is continuous}" shows "A ∈ τ⇩_{1}" proof - from assms have "two_top_spaces1(τ⇩_{1},τ⇩_{2})" and "restrict(f,A):A→X⇩_{2}" and "restrict(f,A) {is continuous}" using tau1_is_top tau2_is_top two_top_spaces1_def fmapAssum restrict_fun by auto then show ?thesis using two_top_spaces1.partial_fun_cont by simp qed text‹If a fuction is continuous on each set of a collection of open sets, then it is continuous on the union of them. We could use continuity with respect to the relative topology here, but we know that on open sets this is the same as the original topology.› lemma (in two_top_spaces0) pasting_lemma1: assumes A1: "M ⊆ τ⇩_{1}" and A2: "∀U∈M. restrict(f,U) {is continuous}" shows "restrict(f,⋃M) {is continuous}" proof - { fix V assume "V∈τ⇩_{2}" from A1 have "⋃M ⊆ X⇩_{1}" by auto then have "restrict(f,⋃M)-``(V) = f-``(V) ∩ (⋃M)" using func1_2_L1 fmapAssum by simp also have "… = ⋃ {f-``(V) ∩ U. U∈M}" by auto finally have "restrict(f,⋃M)-``(V) = ⋃ {f-``(V) ∩ U. U∈M}" by simp moreover have "{f-``(V) ∩ U. U∈M} ∈ Pow(τ⇩_{1})" proof - { fix W assume "W ∈ {f-``(V) ∩ U. U∈M}" then obtain U where "U∈M" and I: "W = f-``(V) ∩ U" by auto with A2 have "restrict(f,U) {is continuous}" by simp with ‹V∈τ⇩_{2}› have "restrict(f,U)-``(V) ∈ τ⇩_{1}" using IsContinuous_def by simp moreover from ‹⋃M ⊆ X⇩_{1}› and ‹U∈M› have "restrict(f,U)-``(V) = f-``(V) ∩ U" using fmapAssum func1_2_L1 by blast ultimately have "f-``(V) ∩ U ∈ τ⇩_{1}" by simp with I have "W ∈ τ⇩_{1}" by simp } then show ?thesis by auto qed then have "⋃{f-``(V) ∩ U. U∈M} ∈ τ⇩_{1}" using tau1_is_top IsATopology_def by auto ultimately have "restrict(f,⋃M)-``(V) ∈ τ⇩_{1}" by simp } then show ?thesis using IsContinuous_def by simp qed text‹If a function is continuous on two sets, then it is continuous on intersection.› lemma (in two_top_spaces0) cont_inter_cont: assumes A1: "A ⊆ X⇩_{1}" "B ⊆ X⇩_{1}" and A2: "restrict(f,A) {is continuous}" "restrict(f,B) {is continuous}" shows "restrict(f,A∩B) {is continuous}" proof - { fix V assume "V∈τ⇩_{2}" with assms have "restrict(f,A)-``(V) = f-``(V) ∩ A" "restrict(f,B)-``(V) = f-``(V) ∩ B" and "restrict(f,A)-``(V) ∈ τ⇩_{1}" and "restrict(f,B)-``(V) ∈ τ⇩_{1}" using func1_2_L1 fmapAssum IsContinuous_def by auto then have "(restrict(f,A)-``(V)) ∩ (restrict(f,B)-``(V)) = f-``(V) ∩ (A∩B)" by auto moreover from A2 ‹V∈τ⇩_{2}› have "restrict(f,A)-``(V) ∈ τ⇩_{1}" and "restrict(f,B)-``(V) ∈ τ⇩_{1}" using IsContinuous_def by auto then have "(restrict(f,A)-``(V)) ∩ (restrict(f,B)-``(V)) ∈ τ⇩_{1}" using tau1_is_top IsATopology_def by simp moreover from A1 have "(A∩B) ⊆ X⇩_{1}" by auto then have "restrict(f,A∩B)-``(V) = f-``(V) ∩ (A∩B)" using func1_2_L1 fmapAssum by simp ultimately have "restrict(f,A∩B)-``(V) ∈ τ⇩_{1}" by simp } then show ?thesis using IsContinuous_def by auto qed text‹The collection of open sets $U$ such that $f$ restricted to $U$ is continuous, is a topology.› theorem (in two_top_spaces0) pasting_theorem: shows "{U ∈ τ⇩_{1}. restrict(f,U) {is continuous}} {is a topology}" proof - let ?T = "{U ∈ τ⇩_{1}. restrict(f,U) {is continuous}}" have "∀M∈Pow(?T). ⋃M ∈ ?T" proof fix M assume "M ∈ Pow(?T)" then have "restrict(f,⋃M) {is continuous}" using pasting_lemma1 by auto with ‹M ∈ Pow(?T)› show "⋃M ∈ ?T" using tau1_is_top IsATopology_def by auto qed moreover have "∀U∈?T.∀V∈?T. U∩V ∈ ?T" using cont_inter_cont tau1_is_top IsATopology_def by auto ultimately show ?thesis using IsATopology_def by simp qed text‹0 is continuous.› corollary (in two_top_spaces0) zero_continuous: shows "0 {is continuous}" proof - let ?T = "{U ∈ τ⇩_{1}. restrict(f,U) {is continuous}}" have "?T {is a topology}" by (rule pasting_theorem) then have "0∈?T" by (rule empty_open) hence "restrict(f,0) {is continuous}" by simp moreover have "restrict(f,0) = 0" by simp ultimately show ?thesis by simp qed end