Developable hyperspaces are metrizable

@ Applied General Topology c Universidad Polit´ecnica de Valencia Volume 4, No. 2, 2003 pp. 351–360 Developable hyperspaces are metrizable L’ubica Hol´ a, Jan Pelant∗and L´ aszl´ o Zsilinszky Dedicated to Professor S. Naimpally on the occasion of his 70th birthday. Abstract. developability of hyperspace topologies (locally finite, (bounded) Vietoris, Fell, respectively) on the nonempty closed sets is characterized. Submetrizability and having a Gδ-diagonal in the hyperspace setting is also discussed. 2000 AMS Classification: 54B20. Keywords: developable spaces, Vietoris topology, Fell topology, locally finite topology, bounded Vietoris topology, Gδ-diagonal. 1. Introduction. Let CL(X) (K(X)) denote the hyperspace of nonempty closed (compact) sets of a T2topological space (X, τ). For notions not defined in the paper see [11], [2] and [15]. Historically there have been two hyperspace topologies of particular importance: the Vietoris topology τV(see section 1) and the Hausdorff metric topology τH, as considered in Michael’s fundamental paper on hyperspaces [22]. It is well-known, that τV=τHon K(X) and hence (K(X), τV) is metrizable iff Xis. Metrizability of the larger hyperspace (CL(X), τV) is also characterized, it is equivalent to Xbeing compact metrizable [22]. To investigate generalized metric properties of the Vietoris topology, one may start by considering Bing’s factorization of metrizability into collectionwise normality (CWN) and Mooreness ([11]). There is an abundance of results on CWN and related properties, e.g. (combined results of Keesling and Velichko from [19], [20], [27]): (CL(X), τV) is CWN (paracompact, normal, resp.) iff X is compact; however, some stronger (hereditary) properties, such as hereditary normality, stratifiability or monotone normality, coincide with metrizability for (CL(X), τV) (cf. [6], [12], [13] and [7]). ∗The second author was supported by the grant GACR 201/00/1466. 352 L’ubica Hol´a, Jan Pelant and L´aszl´o Zsilinszky As far as the other half of Bing’s Theorem is concerned, Mooreness (developability) has been considered only for K(X); indeed, Mizokami has shown that (K(X), τV) is Moore iff Xis [23]. It is one of the purposes of this paper to characterize developability (Mooreness) of the Vietoris topology on CL(X). A good starting point for investigating developability is to look at 1st countability of (CL(X), τV) first, as was done by Hol´a and Levi [16] while extending an older result of Choban [8]: Theorem 1.1. Let Xbe a T2space. The following are equivalent: (i) (CL(X), τV)is 1st countable; (ii) Xis perfectly normal, the derived set X0is countably compact, hereditarily separable and of countable character, and X\X0is countable. Thus, in general, 1st countability of the Vietoris topology does not guarantee its metrizability (just consider X=ω, the discrete space of non-negative integers); other cases, e.g. if Xis dense-in-itself and metrizable, are treated in [10]. In section 2 we will prove that developability and metrizability of (CL(X), τV) always coincide. However, see Remark 3.4 following Theorem 3.3 for a relevant comment. In fact, after obtaining the same relationship for other well-studied hypertopologies, such as the locally finite, Fell and bounded Vietoris topology, respectively (see section 1 for definitions), as well as for the Wijsman topology (which is 1st countable iff it is metrizable [2]), it seems that the coincidence of developability and metrizability in the hyperspace setting could be established for a broad class of hypertopologies. It remains to be seen if it is the case for hit-and-miss and hit-and-far topologies ([2]) or even for the general hyperspace topology studied in [28], incorporating all the above topologies along with some weak hyperspace topologies, including the Wijsman topology. Since metrizability is to submetrizability as developability is to having a Gδdiagonal (see 1.6, 2.2 and 2.5 in [15]), we then turn to studying submetrizability and having a Gδ-diagonal in CL(X). It turns out to be a perfect match for the Fell and the bounded Vietoris topology, respectively, moreover, if Xis Morita’s M-space, also for the Vietoris topology. Finally, in Section 4, the above generalized metric properties are discussed on K(X) with the Vietoris and Fell topologies, respectively. 2. Preliminaries. To describe the hypertopologies we will work with, we need to introduce some notation: for U⊂Xput U+={A∈CL(X) : A⊂U}and U−={A∈CL(X) : A∩U6=∅}. Subbase elements of the Vietoris (locally finite)topology τV(τlf ) on CL(X) are of the form U+with U∈τand TU∈U U−with U ⊂ τfinite (locally finite). Note, that for a metrizable X, the supremum of all Hausdorff metric (resp. Wijsman) topologies corresponding to topologically equivalent metrics is the locally finite (resp. Vietoris) topology ([4], [25], [5]). Developable hyperspaces are metrizable 353 Another classical hypertopology, the Fell topology, has found numerous applications in various fields of mathematics ([21], [1]); it has as a subbase elements of the form U−and V+, where U∈τand Vhas a compact complement in X. If (X, d) is a metric space the bounded Vietoris topology τbVdhas as a subbase, elements of the form U−and V+, where U∈τand the complement of Vis a closed bounded set in (X, d). Proposition 2.1. Let Xbe a T2space. If (CL(X), τV)or (CL(X), τlf )is 1st countable, then Xis collectionwise normal. Proof. In view of Theorem 1.1 (resp. [9]), Xis normal and X0is countably compact; thus, if Dis a discrete family of closed subsets of X, only finitely many members D0, . . . , Dn∈ D intersect X0. Since Xis normal, there exist pairwise disjoint open sets U0, . . . , Unsuch that Di⊂Uifor each i≤n. Denote D0={D0, . . . , Dn}and observe that D=S(D \ D0) is closed. Consequently, (D \ D0)∪ {U0\D, . . . , Un\D}is a disjoint open expansion of D. Proposition 2.2. Let Xbe a T2space. If (CL(X), τV)or (CL(X), τlf )is developable, then Xis metrizable and X0is compact. Proof. By admissibility of the Vietoris and the locally finite topology, Xis a developable space, which is also collectionwise normal by Proposition 2.1; hence, in view of Bing’s Theorem ([11]), Xis metrizable. Since X0is countably compact, in our case, it is compact.  3. Developability in CL(X). Theorem 3.1. Let Xbe a T2space. The following are equivalent: (i) (CL(X), τlf )is Moore; (ii) (CL(X), τlf )is developable; (iii) (CL(X), τlf )is metrizable; (iv) Xis metrizable and X0is compact. Proof. (ii)⇒(iv) follows from Proposition 2.2 and (iv)⇒(iii) from [4], Theorem 2.3.  Theorem 3.2. (CL(ω), τV)is not developable. Proof. Suppose (CL(ω), τV) is developable. Let D={Un:n∈ω}be a development of (CL(ω), τV). Without loss of generality we may suppose that Un+1 is a refinement of Unfor every n. For every A∈[ω]ω, define L(A) = min{n: St(A, Un)⊂A+}, and for every A∈[ω]ωand F∈[A]<ω put π(A, F) = min{L(B) : F⊂B, B is a proper infinite subset of A}. For every A∈[ω]ω,m=L(A), and every F∈[A]<ω choose H(A, F )∈ [A]<ω such that Fis a proper subset of H(A, F ) and there is U∈ Umwith A+∩Tp∈H(A,F ){p}−⊂U. 354 L’ubica Hol´a, Jan Pelant and L´aszl´o Zsilinszky For every A∈[ω]ωand every B, G ∈[A]<ω such that H(A, B)⊂Gwe have π(A, G)> L(A): otherwise, π(A, G)≤L(A) and there is an infinite proper subset Bof Awith G⊂Bsuch that L(B)≤L(A). Hence, there is U∈ UL(A) such that {A, B} ⊂ U. As Un+1 is a refinement of Unfor every n∈ω, we see that for each k < L(A) there is Uk∈ Ukwith {A, B} ⊂ Uk. Hence for every k≤L(A), St(B, Uk) is not subset of B+(as A\B6=∅) and so L(B)> L(A), a contradiction. We use an inductive construction now: •Put n0=π(ω, ∅). Take A0∈[ω]ωsuch that L(A0) = n0and G0∈[ω]<ω such that there is U∈ Un0with A+ 0∩p∈G0{p}−⊂U. Put F0=∅. •Let Fj+1 =Si≤jGiand nj+1 =π(Aj, Fj+1); choose Aj+1 ∈[ω]ωsuch that Fj+1 ⊂Aj+1, where Aj+1 is a proper subset Aj,L(Aj+1) = nj+1 and put Gj+1 =H(Aj+1, Fj+1). Clearly, this construction can be repeated ω-many times. Finally, put B= Si∈ωGiand take p∈ωsuch that np> L(B). Then Fp⊂B⊂Ap−1, but L(B)≥π(Ap−1, Fp) = np, a contradiction.  Theorem 3.3. Let Xbe a T2space. The following are equivalent: (i) (CL(X), τV)is Moore; (ii) (CL(X), τV)is developable; (iii) (CL(X), τV)is metrizable; (iv) Xis compact and metrizable. Proof. Only (ii)⇒(iv) needs justification: in view of Proposition 2.2, it suffices to show that every sequence in X\X0has a cluster point in X. Otherwise, X\X0contains a closed copy of ω, thus, (CL(ω), τV) sits in (CL(X), τV) and is hence developable, a contradiction with Theorem 3.2.  Remark 3.4. After L’. Hol´a’s lecture at Caserta 2001, prof. Arhangel’skiˇı was wondering, whether Theorem 3.3 could be extended to hyperspaces which are σ-spaces (i.e. spaces with a σ-discrete network) (see [15], also for related notions of (strong) Σ-spaces). He was very right. A possible way could be an easy modification of the proof of Theorem 3.2 in effect that (CL(ω), τV) is not aσ-space and an application of the fact that each countably compact σ-space is compact and metrizable [15]. Fortunately, Popov [26] proved already in 1978 the following Theorem 3.5. (CL(ω), τV)contains the Sorgenfrey line Sas a subspace. Recall that Sis not even a Σ-space. So it follows from Popov’s result that if (CL(X), τV) is a Σ-space then Xis countably compact. Of course, Theorem 3.2 represents a special case, nevertheless we have decided to keep its proof to make the paper more self-contained. As M-spaces are Σ-spaces, some other information on the Vietoris hyperspaces which are Σ-spaces, may be found in Developable hyperspaces are metrizable 355 Proposition 4.9 below. Coming back to the original Arhangel’skiˇı’s question, we could reformulate Theorem 3.3: Theorem 3.6. Let Xbe a T2space. The following are equivalent: (o) (CL(X), τV)has a σ-discrete network; (i) (CL(X), τV)is Moore; (ii) (CL(X), τV)is developable; (iii) (CL(X), τV)is metrizable; (iv) Xis compact and metrizable. We proceed with other topologies now: Theorem 3.7. Let (X, d)be a metric space. Then the following are equivalent: (i) (CL(X), τbVd)is Moore; (ii) (CL(X), τbVd)is developable; (iii) (CL(X), τbVd)is metrizable; (iv) (X, d)is boundedly compact (i.e. every closed bounded set in (X, d)is compact). Proof. Since (iii)⇔(iv) is known, only (ii)⇒(iv) needs some comments: let B∈CL(X) be bounded in (X, d). Developability of (CL(X), τbVd) implies that CL(B) equipped with the relative topology τbVdon CL(B) is also developable. It is easy to verify that the relative topology τbVdon CL(B) coincides with the Vietoris topology τV. Thus, (CL(B), τV) is developable and Bmust be compact by Theorem 3.3.  Theorem 3.8. Let Xbe a T2space. The following are equivalent: (i) (CL(X), τF)is Moore; (ii) (CL(X), τF)is developable; (iii) (CL(X), τF)is T2and has a Gδ-diagonal; (iv) (CL(X), τF)is submetrizable; (v) (CL(X), τF)is metrizable; (vi) Xis hemicompact and metrizable. Proof. (ii)⇒(iii) Developability of (CL(X), τF) implies that it has a Gδ-diagonal; moreover, even 1st countability of (CL(X), τF) implies, that Xis locally compact ([16]), so, (CL(X), τF) is T2by a result of Fell. (iii)⇒(v) Hausdorffness of (CL(X), τF) implies that (CL(X), τF) is locally compact. Since (CL(X), τF) has a Gδ-diagonal, points of (CL(X), τF) are Gδ; thus, (CL(X), τF) is 1st countable; hence (see [17]) it is paracompact. In summary, (CL(X), τF) is paracompact, locally compact with a Gδ-diagonal and is therefore metrizable. (v)⇔(vi) is known ([3]) (to prove (vi)⇒(v) realize that every first countable hemicompact Hausdorff space is locally compact). The remaining implications are trivial.  356 L’ubica Hol´a, Jan Pelant and L´aszl´o Zsilinszky 4. Submetrizability, having a Gδ-diagonal and related properties in CL(X). The last theorem of the previous section showed that these properties coincide for (CL(X), τF). In what follows, we show that similar relationship holds for the (bounded) Vietoris topology as well. First, we will see that for the (bounded) Vietoris and locally finite topology, respectively, submetrizability and developability are distinct. Proposition 4.1. (i) If Xis a metrizable space, then (CL(X), τlf )is submetrizable. (ii) If (X, d)is a separable metric space, then (CL(X), τV)and (CL(X), τbVd) are submetrizable. Proof. (i) If Xis a metrizable space, take any compatible metric don Xand consider the Hausdorff metric topology τHd. It is known that τHd⊆τlf . (ii) Since dis a separable metric on X, the Wijsman topology τWdis metrizable ([2]). It is known that τWd⊆τbVd⊆τV. Corollary 4.2. (i) (CL(X), τV)is submetrizable and not developable, if Xis non-compact, separable and metrizable. (ii) (CL(X), τbVd)is submetrizable and not developable, if (X, d)is a separable metric space which is not boundedly compact. (iii) (CL(X), τlf )is submetrizable and not developable, if Xis a non-compact dense-in-itself metrizable space. Proposition 4.3. Let Xbe a T2space. (i) If the points in (CL(X), τV)are Gδ, then Xis hereditarily separable and every closed set in Xis Gδ. (ii) If the points in (CL(X), τF)are Gδ, then Xis hereditarily separable, every open set in Xis σ-compact and every closed set in Xis a Gδ-set. (iii) If (X, d)is a metric space, then the points in (CL(X), τbVd)are Gδiff (X, d)is separable. Proof. We prove only (i): let A⊂X. Since Ais a Gδ-set in (CL(X), τV), there are τV-open sets Gnsuch that {A}=Tn∈ωGn. Without loss of generality we can suppose, that for every n∈ω,Gn= (Gn)+∩∩l≤n(Uil n)−, where Gn, Uil n, l ≤ nare open sets in X. For every n∈ω,l≤nchoose ail n∈A∩Uil n. It is easy to verify that {ail n:l≤n, n ∈ω}=A. Remark 4.4. Let Xbe a T2space. (i) If (CL(X), τV) has a Gδ-diagonal, then Xis hereditarily separable. (ii) Points in (CL(X), τV) are Gδiff every A∈CL(X) is a Gδ-set and has a countable pseudobase (in A). Proposition 4.5. Let (X, d)be a metric space. The following are equivalent: (i) (CL(X), τbVd)is submetrizable; Developable hyperspaces are metrizable 357 (ii) (CL(X), τbVd)has a Gδ-diagonal; (iii) (X, d)is separable. Proof. (iii)⇒(i) See Proposition 4.1(ii). (ii)⇒(iii) Follows from Proposition 4.3(iii).  Proposition 4.6. Let Xbe a w∆-space. The following are equivalent: (i) (CL(X), τV)is submetrizable; (ii) Xis a separable metrizable space. Proof. (ii)⇒(i) See Proposition 4.1(ii). (i)⇒(ii) Submetrizability of (CL(X), τV) implies its Hausdorffness, so Xis regular by a result of Michael [22] and submetrizable, which in turn, being a w∆-space, is an M-space ([15]). However, an M-space with a Gδ-diagonal is metrizable ([15]). Finally, by Remark 4.4(i), Xis separable.  Proposition 4.7. Let Xbe an M-space. The following are equivalent: (i) (CL(X), τV)is submetrizable; (ii) (CL(X), τV)is T2and has a Gδ-diagonal. (iii) Xis a separable metrizable space. Proof. ((ii)⇒(i) See Proposition 4.1(ii). (i)⇒(ii) An M-space with a Gδ-diagonal is metrizable ([15]) and by Remark 4.4(ii), Xis separable.  Proposition 4.8. Let Xbe a T2space. (i) (CL(X), τV)is a w∆-space (strict p-space,M-space, resp.) ⇒Xis countably compact. (ii) Xis countably compact ;(CL(X), τV)is a w∆-space. Proof. (i) By Proposition 4.1(ii), (CL(ω), τV) is submetrizable, so it has a G∗ δdiagonal ([15]). By a theorem of Hodel ([15]) this implies, that (CL(ω), τV) is not a w∆-space (neither is a strict p-space or an M-space, since these properties are stronger than w∆). On the other hand, if (CL(X), τV) is a w∆-space and X is not countably compact, then ωsits in Xas a closed subset and hence CL(ω) embeds as a closed subset in (CL(X), τV). Since the w∆-property is closed hereditary, this would imply that (CL(ω), τV) is a w∆-space, a contradiction. (ii) By Example 2 of [24], there is a countably compact space X, such that (F2(X), τV) (= the space of all sets with at most 2 elements) is not a w∆- space. Since F2(X) is a closed set in (CL(X), τV) and the w∆-property is closed hereditary, (CL(X), τV) is not a w∆-space.  A topological space Xis ultracompact iff every net in Xwith a countable range has a cluster point. This characterization of ultracompactness is due to Hol´a and K¨unzi [18]. Proposition 4.9. Let Xbe a linearly ordered topological space. The following are equivalent: (o) (CL(X), τV)is countably compact; 358 L’ubica Hol´a, Jan Pelant and L´aszl´o Zsilinszky (i) (CL(X), τV)is an M-space; (ii) (CL(X), τV)is a w∆-space; (iii) Xis countably compact. Proof. (ii)⇒(iii) See Proposition 4.8(i). For the rest of the proof, realize that in linearly ordered topological spaces countable compactness and ultracompactness coincide [14], and Xis ultracompact iff (CL(X), τV) is [18]. Further, an ultracompact space is countably compact, which in turn is an M-space.  Remark 4.10. It can be inferred from the above proof, that if Xis ultracompact, then (CL(X), τV) is an M-space (w∆-space). So, for every non-compact ultracompact space X, (CL(X), τV) is a non-compact M-space (w∆-space). Proposition 4.8 also offers another proof of developability of (CL(X), τV): Proposition 4.11. Let Xbe a T2space. The following are equivalent: (i) (CL(X), τV)is developable; (ii) (CL(X), τV)is a w∆-space and Xhas a Gδ-diagonal. Proof. (i)⇒(ii) is clear. (ii)⇒(i) By Proposition 4.8(i), Xis countably compact. Every countably compact space with a Gδ-diagonal is compact and metrizable [15]. Now Theorem 3.3 applies.  5. Some generalized metric properties in K(X). Proposition 5.1. Let Xbe a T2space. The following are equivalent: (i) (K(X), τV)is submetrizable; (ii) Xis submetrizable. Proof. (i)⇒(ii) It is easy to verify that if Tis a coarser metrizable topology on (K(X), τV), then the relative topology Ton Xis coarser than τ. (ii)⇒(i) Let µ⊆τbe a metrizable topology. Then K(X, τ) (= compact sets in τ)⊆K(X, µ). (K(X, µ), µV) is a metrizable space, where µVis the Vietoris topology generated by µ. Thus, also µVrestricted to K(X, τ) is a metrizable topology on K(X, τ). Since µ⊆τ, we have that µVrestricted on K(X, τ) is coarser than τV. The situation in (K(X), τF) is different: Proposition 5.2. Let Xbe a T2space. The following are equivalent: (i) (K(X), τF)is Moore; (ii) (K(X), τF)is developable; (iii) (K(X), τF)is T2and has a Gδ-diagonal; (iv) (K(X), τF)is submetrizable; (v) (K(X), τF)is metrizable; (vi) Xis hemicompact and metrizable. Developable hyperspaces are metrizable 359 Proof. (iii)⇒(vi) First we show that if (K(X), τF) is T2, then Xis locally compact. Suppose that Xfails to be locally compact. Let x∈Xbe such that for every U∈ B(x) and K∈K(X) there is some xU,K ∈U\K(here B(x) stands for a base of neighborhoods of x). Let y∈Xbe a point different from x. It is easy to verify that the net of compact sets {{xU,K , y}:U∈ B(x), K ∈K(X)} converges both to {x, y}and to {y}in (K(X), τF), a contradiction. Since (K(X), τF) has a Gδ-diagonal, points of (K(X), τF) are Gδand Xhas aGδ-diagonal. Thus, Xis σ-compact, locally compact with a Gδ-diagonal, i.e. it must be hemicompact, by [11] 3.8.C (b), and metrizable. (vi)⇒(v) If Xis hemicompact and metrizable, then (CL(X), τF) is metrizable [2]; thus, (K(X), τF) is also metrizable. (v)⇒(iv), (iv)⇒(iii) and (i)⇒(ii) are trivial. (ii)⇒(iii) It follows from [3], that 1st countability of (K(X), τF) implies hemicompactness of Xand hence its local compactness (since Xmust be first countable). This in turn is equivalent to Hausdorffness of (K(X), τF). 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