Baireness of the Wijsman hyperspace topology is characterized for a metrizable base space with a countable-in-itself p-base; further, a separable 1st category metric spaceis constructed with a Baire Wijsman hyperspac

NC DOCKS at The University of North Carolina at Pembroke

Zsilinszky, Laszlo

English

The University of North Carolina at Greensboro

ON BAIRENESS OF THE WIJSMAN HYPERSPACELÁSZLÓ ZSILINSZKYAbstract. Baireness of the Wijsman hyperspace topology is characterized for a metrizablebase space with a countable-in-itself π-base; further, a separable 1st category metric spaceis constructed with a Baire Wijsman hyperspace.1. IntroductionThere has been a considerable effort in exploring various completeness properties of theWijsman hyperspace topology wd, i.e. the weak topology on the nonempty closed subsetsof the metric space (X, d) generated by the distance functionals viewed as functions of setargument [17]. It was first shown by Effros [10], that a Polish space admits a metric for whichthe Wijsman topology is Polish; later, Beer showed [2],[3], that given a separable completemetric base space, the corresponding Wijsman hyperspace is Polish. Finally, Costantinidemonstrated in [6], that Polish base spaces always generate Polish Wijsman topologies (ashort proof, using the so-called strong Choquet game, was found by the author in [19]). Asa related result, note that the Wijsman hyperspace is analytic iff X is analytic [1].Beer asked, whether complete metrizability ofX alone (without separability) is equivalentto some completeness property of the Wijsman hyperspace. Costantini [7] showed that anatural candidate, Čech-completeness, is not the right property; on the other side, completemetrizability of X guarantees Baireness [18], even strong α-favorability [19], of the Wijsmanhyperspace regardless of the underlying metric on X. It is also known, that less thancomplete metrizability ofX - e.g. having a dense completely metrizable subspace [20] or beinga separable Baire space [18], respectively - guarantees Baireness of the Wijsman topology;however, wd may be non-hereditarily Baire, even if X is separable, hereditarily Baire and hasa dense completely metrizable subspace [20], or X is completely metrizable [9], respectively.It is the purpose of this paper to continue in this research by characterizing Baireness ofthe Wijsman hyperspace for almost locally separable metrizable spaces.A space is almost locally separable, provided the set of points of local separability isdense. In a metrizable space, this is equivalent to having a countable-in-itself π-base, i.e. aπ-base, each element of which contains only countably many elements of the π-base [21] (cf.locally countable pseudo-base of Oxtoby [14]). A topological space is a Baire space, provided2000 Mathematics Subject Classification. Primary 54B20; Secondary 54E52.Key words and phrases. Wijsman topology, ball (proximal) topology, Baire space, countable-in-itself π-base.12 LÁSZLÓ ZSILINSZKYcountable collections of dense open subsets have a dense intersection or, equivalently, ifnonempty open sets are of 2nd Baire category [12].Let CL(X) stand for the space of nonempty closed subsets of a topological space (X, τ)(the so-called hyperspace of X), and for M ⊆ X defineM− = {A ∈ CL(X) : A ∩M 6= ∅},M+ = {A ∈ CL(X) : A ⊆M}.We will write M c, M for the complement and closure, respectively, of M in X. If (X, τ) isa metrizable topological space with a compatible metric d, denote by S(x, ε) (resp. B(x, ε))the open (resp. closed) ball of radius ε > 0 about x ∈ X, and put S(M, ε) =⋃m∈M S(m, ε)for the open ε-hull of M ⊆ X. Denote by B(X) the collection of finite unions of closed balls.The Wijsman topology wd on CL(X) is the weak topology generated by the distancefunctionals d(x,A) = inf{d(x, a); a ∈ A} (x ∈ X,A ∈ CL(X)) viewed as functionals ofset argument. It is easy to show that subbase elements of wd are of the form U− and{A ∈ CL(X) : d(x,A) > ε}, where U ∈ τ, x ∈ X and ε > 0. The Wijsman topology is afundamental tool in the construction of the lattice of hyperspace topologies, since many ofthese arise as suprema and infima, respectively of appropriate Wijsman topologies [4],[8].The ball proximal topology bpd has subbase elements of the form U− and {A ∈ CL(X) :inf{d(a, b) : (a, b) ∈ A × B} > ε}, where U ∈ τ , B ∈ ∆, ε > 0; it coincides with theWijsman topology when X is a normed space (for a characterization of this coincidence see[11]). Moreover, bpd is Baire if and only if wd is [18]. As we will see, there is an even simplerhypertopology on CL(X) with this “Baire connection”, the so-called ball topology.The ball topology bd has subbase elements of the form U− and (Bc)+, where U ∈ τ andB ∈ B(X). It is not hard to show that the collectionB = {(Bc)+ ∩⋂i≤nS(xi, r)− : B ∈ B(X), r > 0, n ∈ ω,the S(xi, r)’s are pairwise disjoint and miss B}is a base for bd. The ball topology is a hit-and-miss topology, like the well-known Vietorisor Fell topologies [3]; bd may be non-regular [11], so it is certainly not always equal to the(completely regular) Wijsman topology, however, we have:Theorem 1.1. The following are equivalent:(i) (CL(X), wd) is a Baire space,(ii) (CL(X), bd) is a Baire space.Proof. If Uw =⋂i≤n U−i ∩⋂j≤m{A ∈ CL(X) : d(xj, A) > εj} ∈ wd is nonempty, r =min{d(xj ,A)−εj2: j ≤ m}, andUb =⋂i≤nU−i ∩⋂j≤m((B(xj, εj + r))c)+ ∈ bd,then ∅ 6= Ub ⊆ Uw.ON BAIRENESS OF THE WIJSMAN HYPERSPACE 3Conversely, if Ub =⋂i≤n U−i ∩⋂j≤m((B(xj, εj))c)+ ∈ bd is nonempty, and ai ∈ Ui so thatd(xj, ai) > εj for all i, j, then r = mini,j d(xj, ai) > εj for all j, and forUw =⋂i≤nU−i ∩⋂j≤m{A ∈ CL(X) : d(xj, A) >εj + r2} ∈ wdwe have {a0, . . . , an} ∈ Uw ⊆ Ub. The theorem now follows by [16], Proposition 3.4.  2. Main ResultsWe need some auxiliary material first: natural numbers will be viewed as sets of pre-decessors, ∆ ⊆ CL(X) will be closed under finite unions, bold symbols will denote notionsrelated to the product space X = Xω endowed with the so-called pinched-cube topology [15]τ = τ (∆) having the baseB = {(∏i≤nUi)× (Bc)ω\(n+1) : B ∈ ∆, n ∈ ω, Ui ∈ τ, Ui ⊆ Bc ∀ i ≤ n}.If J ⊆ ω, the projection map πJ : (X, τ ) → XJ is continuous and open. If C ⊆ X, J ⊆ ωand x ∈ XJ , denote C[x] = C ∩ π−1J (x); further, if C is a collection of subsets of X, putC[x] = {C ∈ C : C[x] 6= ∅}.The proof of the next theorem is a modification of analogous results about the Tychonoffand box products of Baire spaces from [21], we sketch the proof for completeness:Theorem 2.1. If (X, τ) is a topological Baire space with a countable-in-itself π-base, then(X, τ ) is a Baire space.Proof. Let P be a countable-in-itself π-base of X. Let {Gn}n be a decreasing sequenceof dense open subsets of (X, τ ) and fix a nonempty τ -open V. Choose V0 ∈ B withV0 ⊆ V ∩ G0, π1(V0) ∈ P and put B0 = {V0}. By induction, we can define Bi ⊆ Bfor each i ≥ 1 so that Bi =⋃B∈Bi−1 Bi(B), where for all B ∈ Bi−1, Bi(B) is a maximalcollection such that(1) A ⊆ B ∩Gi for each A ∈ Bi(B),(2) πi+1(A) ∈ {∏k≤i Pk : (P0, . . . , Pi) ∈ Pi+1} for each A ∈ Bi(B),(3) {πi(A) : A ∈ Bi(B)} is pairwise disjoint.Notice, that each Bi(B) is countable, since, by (2), πi(B) has ccc for each i ≥ 1 andB ∈ Bi−1. For i ≥ 1 denoteUi =⋃B∈Bi{U ∈ B : πω\i(U) = πω\i(B) and U ⊆ B}.For each B ∈ B1(V0) defineY(0)B = {x ∈ X : B[x] 6= ∅ and ∀U ∈ U2(U ⊆ B⇒ U[x] = ∅)}.Claim 2.1.1. Y(0)B is nowhere dense in X for each B ∈ B1(V0).4 LÁSZLÓ ZSILINSZKY(Assume by contradiction that for some B ∈ B1(V0), Y (0)B is dense in some U ∈ τ .Then B ∩ π−11 (U) 6= ∅, and by maximality of B2(B), there exists E ∈ B2(B) such thatU = E ∩ π−11 (U) 6= ∅. Then U ∈ U2, U ⊆ B, and π1(U) is a nonempty open subset of U ;thus, it intersects with Y(0)B , say, in x. To get a contradiction, note that x ∈ π1(U) meansU[x] 6= ∅; on the other hand, x ∈ Y (0)B implies U[x] = ∅.)Define W0 = π1(⋃B1(V0)). Since X is a Baire space, by Claim 2.1.1, we can choosesomex0 ∈ W0 \⋃{Y (0)B : B ∈ B1(V0)}.Then, by (3), there is a unique V1 ∈ B1(V0)[x0]; further, for each B ∈ B1(V0)[x0] there isa U ∈ U2 with U ⊆ B and U[x0] 6= ∅.By induction, assume that for all i ≤ n (n ∈ ω), Vi ∈ Bi, andxi ∈ Wi = π(i+1)\i(⋃Bi+1(Vi)[x0, . . . , xi−1])have been defined (when i = 0, B1(V0)[x0, x−1]) is meant to be B1(V0)) so that(4) ∀1 ≤ i ≤ n (Vi ∈ Bi(Vi−1)[x0, . . . , xi−1]),(5) ∀i ≤ n∀B ∈ Bi+1(Vi)[x0, . . . , xi] ∃U ∈ Ui+2 (U ⊆ B and U[x0, . . . , xi] 6= ∅).Since xn ∈ Wn, and, by (3), {π(n+1)\n(A) : A ∈ Bn+1(Vn)[x0, . . . , xn−1]} is pairwise disjoint,there is a unique Vn+1 ∈ Bn+1(Vn)[x0, . . . , xn−1] with Vn+1[x0, . . . , xn] 6= ∅; thus, (4) holdsfor 1 ≤ i ≤ n+ 1.Moreover, (5) implies that there is some U ∈ Un+2 with U ⊆ Vn+1 and U[x0, . . . , xn] 6=∅. We can also find B ∈ Bn+2 with U ⊆ B and πω\(n+2)(U) = πω\(n+2)(B). By (1),B ⊆ Gn+2, and since B intersects Vn+1, it follows by (3), that B ⊆ Vn+1; thus, B ∈Bn+2(Vn+1)[x0, . . . , xn], soWn+1 = π(n+2)\(n+1)(⋃Bn+2(Vn+1)[x0, . . . , xn])is a nonempty open subset of X. For each B ∈ Bn+2(Vn+1)[x0, . . . , xn] defineY(n+1)B = {x ∈ X : B[x0, . . . , xn, x] 6= ∅ and∀U ∈ Un+3(U ⊆ B⇒ U[x0, . . . , xn, x] = ∅)}.Claim 2.1.2. Y(n+1)B is nowhere dense in X for each B ∈ Bn+2(Vn+1)[x0, . . . , xn].(Indeed, if Y(n+1)B is dense in some U ∈ τ , then B∩π−1(n+2)\(n+1)(U) 6= ∅, and by maximalityof Bn+3(B), there exists E ∈ Bn+3(B) such that U = E ∩ π−1(n+2)\(n+1)(U) 6= ∅. ThenU ∈ Un+3, U ⊆ B, and π(n+2)\(n+1)(U) is a nonempty open subset of U ; thus, it intersectswith Y(n+1)B , say, in x. Finally, x ∈ π(n+2)\(n+1)(U) means that U[x] 6= ∅; on the other hand,x ∈ Y (n+1)B implies U[x] = ∅.)Now, X is a Baire space so, by Claim 2.1.2, we can findxn+1 ∈ Wn+1 \⋃{Y (n+1)B : B ∈ Bn+2(Vn+1)[x0, . . . , xn]},ON BAIRENESS OF THE WIJSMAN HYPERSPACE 5so (5) is satisfied for i ≤ n + 1. By induction, we have constructed x = (xn)n∈ω ∈ X, anda sequence {Vn ∈ B : n ∈ ω} with Vn+1 ∈ Bn+1(Vn)[x0, . . . , xn] for all n ∈ ω. Thenx ∈ Vn ⊆ V ∩Gn for each n ∈ ω, so V ∩⋂n Gn 6= ∅, and (X, τ ) is a Baire space.  Theorem 2.2. Let X be metrizable with a compatible metric d.If (X, τ (B(X))) is a Baire space, then (CL(X), bd) is a Baire space.Proof. The setS(X) = {A ∈ CL(X) : A separable}is dense in (CL(X), bd), since even the set of finite subsets of X is; thus, we only need toprove that (S(X), bd S(X)) is a Baire space. Define ϕ : (X, τ )→ (S(X), bd S(X)) viaϕ(x) = {xk : k ∈ ω}, where x = (xk)k ∈ X.We will be done, if we show that ϕ is continuous and feebly open (i.e. the interior of ϕ(V)is nonempty for each nonempty V ∈ τ ), since Baire spaces are invariant of these mappings(see [12] or [13]).To see continuity, take a basic V = (Bc)+ ∩⋂i≤m V−i ∈ B, where B =⋃j≤pB(zj, εj) ∈B(X). If x = (xk)k ∈ ϕ−1(V), then ϕ(x) ∈ V , so there exists a ki ∈ ω with xki ∈ Vi foreach i ≤ m. We can find some δj > εj so that d(zj, f(x)) > δj for each j ≤ p, as well asneighborhoods Ui of xki (for all i ≤ m) that are subsets of U = (⋃j≤pB(zj, δj))c. ThenU = (∏i≤m Ui)× Uω\(m+1) ∈ B, and x ∈ U ⊆ ϕ−1(V).To justify feeble openness of ϕ, take a nonempty V = (∏i≤m Vi)×(Bc)ω\(m+1) ∈ B, whereB =⋃j≤pB(zj, εj) ∈ B(X). Denote V = (Bc)+ ∩⋂i≤m V−i ∈ B, and take an A ∈ V ∩S(X).If C = {ck : k ∈ ω} is a countable dense subset of A, where ci ∈ C ∩ Vi for each i ≤ m,then c = (ck)k ∈ V, so A = ϕ(c) ∈ ϕ(V). Consequently, ∅ 6= V ∩ S(X) ⊆ ϕ(V).  Theorem 2.3. If X is almost locally separable, then the following are equivalent:(i) (CL(X), wd) is Baire for each compatible metric d on X,(ii) (CL(X), bpd) is Baire for every compatible metric d on X,(iii) (CL(X), bd) is Baire for every compatible metric d on X,(iv) X is a Baire space.Proof. (i)⇔(ii)⇔(iii) follows from Theorem 1.1, and (iv)⇒(iii) from Theorems 2.1 and 2.2(iii)⇒(iv) If X is not a Baire space, it has a nonempty open 1st category subset U , andso there exists a separable closed 1st category subset C of U with a nonempty interior intC.If d0 is a compatible totally bounded metric on C, then by a theorem of Bing [5], it can beextended to a compatible metric d on X. Let Cn be an increasing sequence of closed nowheredense sets such that C =⋃n∈ω Cn. By total boundedness of d on C, there exists a finite setFk,n ⊆ Cn for each n ∈ ω, k ≥ 1, such that Cn ⊆ S(Fk,n, 1k ). DefineGn =⋃k≥1⋃x∈Fk,n(B(x, 1k)c)+.6 LÁSZLÓ ZSILINSZKYThen Gn is clearly bd-open. Moreover, it is also dense in (CL(X), bd): indeed, let V =(Bc)+ ∩⋂i≤m V−i ∈ B. For each i ≤ m there exist ki ≥ 1 and vi ∈ Vi \ S(Cn, 1ki ), sinceotherwise, Vi ⊆⋂k≥1 S(Cn,1k) = Cn for some i ≤ m, which would contradict nowheredensity of Cn. If k = max{ki : i ≤ m} and A = {v0, . . . , vm}, then A ∩ S(Fk,n, 1k ) = ∅, soA ∈ V ∩ Gn. To conclude, notice that⋂n∈ω Gn is disjoint to (intC)−.  Since the Wijsman hyperspace is metrizable iff X is separable [3], we haveCorollary 2.4. The following are equivalent:(i) (CL(X), wd) is a metrizable Baire space for each compatible metric d on X,(ii) X is a separable Baire space.In light of Theorem 2.3, it is natural to ask whether Baireness of just a single Wijsmantopology wd (bpd, bd, respectively) implies Baireness of X. The following example (given byR. Pol) shows, that this is not the case:Example 2.5. There exists a separable 1st category metric space with a Baire Wijsman (ballproximal, ball, resp.) hyperspace.Proof. Consider ωω with the Baire metrice(x, y) = 1/min{n : x(n) 6= y(n)}and its 1st category subset ω<ω of sequences eventually equal to zero. Then the product X =ω<ω × ωω is a separable, 1st category space endowed with the metric d((x0, x1), (y0, y1)) =max{e(x0, y0), e(x1, y1))}.We claim that (CL(X), bd) is a Baire space: let p : X → ω<ω be the projection onto thefirst axis. Let G1 ⊇ G2 ⊇ . . . be dense open sets in (CL(X), bd), and U ∈ B. Inductively, wewill define Ui ∈ B with Ui ⊆ Gi, and finite sets Fi ∈ Ui such that for each u ∈ Fi there isu? ∈ Fi+1 with p(u) = p(u?) and d(u, u?) < 1i .Let U1 ∈ B be a nonempty subset of G1, and choose a finite set F1 ∈ U1. Suppose thatUi and Fi have been defined for some i ≥ 1. Ifmi ≥ max{n : p(u)(n) 6= 0 for some u ∈ Fi},and u′ ∈ S(u, 1mi) for some u ∈ Fi, then p(u) = p(u′). Since Gi+1 is dense and Fi ∈Ui ∩⋂u∈Fi S(u,1mi), we can find Ui+1 ∈ B withUi+1 ⊆ Gi+1 ∩ Ui ∩⋂u∈FiS(u, 1mi),and choose a finite Fi+1 ∈ Ui+1. Assume thatUi = (Bci )+ ∩⋂u∈FiV −u ∈ B.For any u ∈ Fi, the sequence u, u?, u??, . . . is Cauchy in {p(u)}×ωω, so it converges to someu∞ ∈ S(u, 1mi) ⊆ Vu. By the definition of B, the set {u∞ : u ∈⋃i≥1 Fi} misses the clopen BiON BAIRENESS OF THE WIJSMAN HYPERSPACE 7for each i ≥ 1, so∅ 6= {u∞ : u ∈⋃i≥1Fi)} ∈⋂i≥1Ui ⊆ U ∩⋂i≥1Gi;thus, (CL(X), bd) is a Baire space.  Acknowledgements. The author wishes to thank Roman Pol for communicating Ex-ample 2.5 to him, as well as the referee for the useful comments concerning Theorem 2.1that greatly improved the paper.References1. A. Barbati, The hyperspace of an analytic metrizable space is analytic, Rend. Ist. Mat. Univ. Trieste 25(1993), 15–21.2. G. Beer, A Polish topology for the closed subsets of a Polish space, Proc. Amer. Math. Soc. 113 (1991),1123-1133.3. G. Beer, Topologies on Closed and Closed Convex Sets, Kluwer, Dordrecht, 1993.4. G. Beer, A. Lechicki, S. Levi and S. 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Math. 183 (2004), 115–121.Department of Mathematics and Computer Science, University of North Carolina atPembroke, Pembroke, NC 28372, USAE-mail address: laszlo@uncp.edu

On Bairness of the Wijsman Hyperspace

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On Bairness of the Wijsman Hyperspace

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