A homeomorphism induces a natural bijection between the family of open sets. Weconsider the inverse problem and obtain an answer and counter examples

Ito  Yoshinori

Uchida  Fuichi

Yamagata University Academic Repository

Bull. of Yamagata Univ., Nat. Sci., Vol.15, No.3, Feb. 2003Bijections between topologies∗Yoshinori ITOˆ† and Fuichi UCHIDA‡(Received May 21, 2002)AbstractA homeomorphism induces a natural bijection between the familyof open sets. We consider the inverse problem and obtain an answerand counter examples.0 IntroductionLet X and Y be topological spaces, f : X → Y a homeomorphism. Then,f induces a natural bijection f∗ : O(X) → O(Y ), where O( ) denotes thefamily of open sets of the topological space. Indeed, f∗(U) is to be the imagef(U) for any open set U of X. Notice that the bijection f∗ preserves theinclusion relation, that is, U ⊂ V ⇐⇒ f∗(U) ⊂ f∗(V ).Now we consider the inverse problem : “Which bijection ψ : O(X) →O(Y ) is induced by a homeomorphism ?”Our answer is the following. “Assume the topologies O(X) and O(Y )satisfy the T1-axiom, that is, each one-point set is closed. If a bijectionψ : O(X) → O(Y ) preserves the inclusion relation, then ψ is induced by ahomeomorphism f : X → Y .”Next, we will give countably many topologies on the set of natural num-bers such that between any two of them, there is a bijection which preservesthe inclusion relation but it is not induced by any homeomorphism.∗2000 AMS Mathematics Subject Classification. Primary 54A10†Department of Mathematical Sciences, Faculty of Science,Yamagata University, Yama-gata 990-8560, Japan (e-mail address : y.simon-1978-119-41@ezweb.ne.jp)‡Department of Mathematical Sciences, Faculty of Science,Yamagata University, Yama-gata 990-8560, Japan (e-mail address : fuchida@sci.kj.yamagata-u.ac.jp)45Yoshinori ITOˆ and Fuichi UCHIDA1 Main resultTheorem Let (X,O(X)) and (Y,O(Y )) be T1-topological spaces. Letψ : O(X) → O(Y ) be a bijection which preserves the inclusion relation.Then, ψ is induced by a homeomorphism f : X → Y .Proof. Let ψ : O(X)→ O(Y ) be a bijection which preserves the inclusionrelation. Then, we easily see the following equations :ψ(X) = Y, ψ(U ∩ V ) = ψ(U) ∩ ψ(V ),ψ(φ) = φ, ψ(∪αUα) = ∪αψ(Uα).Moreover, we assume the topologiesO(X) andO(Y ) satisfy the T1-axiom.Then, we see that each x ∈ X corresponds uniquely to an element y ∈ Ysatisfying the relation :ψ({x}c) = {y}c.Here, {x}c denotes the complement of the one-point set {x}.The above relation induces a mapping f : X → Y by ψ({x}c) = {f(x)}c.By the assumption, we see f is a bijection. Moreover, we see the following :x ∈ U ⇐⇒ f(x) ∈ ψ(U).Hence we seef(U) := {f(x) | x ∈ U} = ψ(U)for any U ∈ O(X). The equation shows that f is an open map andf−1(V ) = ψ−1(V )for any V ∈ O(Y ). The last equation shows that f is continuous. And hencef : X → Y is a homeomorphism.Therefore, we see ψ = f∗, that is, ψ is induced by the homeomorphismf : X → Y . q.e.d.2 Counter examplesDenote by N the set of all natural numbers. PutUt = {n ∈ N | n > t}, (t = 1, 2, 3, · · · ).DefineOp = {N, φ} ∪ {Ut | t ≥ p}, (p = 1, 2, 3, · · · ).46Bijections between topologiesThen we see Op (p = 1, 2, 3, · · · ) is a topology on N.Let p, q be natural numbers such that p ≤ q. Then, the bijection ψ :Op → Oq is defined by ψ(Ut) = Ut+q−p. Clearly, ψ preserves the inclusionrelation.On the other hand, there exists the smallest non-empty closed subset inthe topological space (N,Op), which is the set {1, 2, · · · , p} of p elements.Therefore, the topological space (N,Op) is not homeomorphic to the topo-logical space (N,Oq) for any p = q, and hence the bijection ψ : Op → Oq isnot induced by any homeomorphism, if p = q.References[1] N.Bourbaki : Elements of Mathematics, General Topology, Chapters 1-4,Springer-Verlag, 2nd printing 1989.[2] J.L.Kelley : General Topology, Graduate Texts in Mathematics 27,Springer-Verlag 1955.47

Bijections between topologies

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Bijections between topologies

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