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On the Dissonance of Some Metrizable Spaces View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector TOPOLOGY AND ITS APPLICATIONS ELSEVIER Topology and its Applications 84 (1998) 259-268 On the dissonance of some metrizable spaces C. Costantinia>*, S. Watsonb a Dipartimento di Matematica, Universitri di Torino, Via Carlo Albert0 10, 10123 Torino. Italy b Department of Mathematics and Statistics. York University 4700 K&Q Street, North York, Ontario. Canada M3J lP3 Received 21 March 1996; revised 7 October 1996 Abstract We prove that for every separable, O-dimensional me&able space X without isolated points, such that every compact subset of it is scattered, the cocompact topology on the hyperspace of X does not coincide with the upper Kuratowski topology-that is, X is dissonant. In particular, it follows that the rational line is dissonant, and that there exist dissonant, hereditarily Baire, separable metrizable spaces. 0 1998 Published by Elsevier Science B.V. Keywords: Hyperspace; Cocompact topology; Upper Kuratowski convergence; Sequential space; Separable metrizable space: O-dimensional space; Scattered compact space AMS classijcation: Primary 54B20, Secondary 54E35; 54D55: 54G12 0. Introduction Since the first appearance of hyperspace topologies, one of the most natural and well- studied problems has been that of their mutual relationships, and in particular of finding necessary and sufficient conditions on the base space for two given hypertopologies to coincide. In this vein, the paper [2] and the recent book of Beer [l] collect a large number of results, and other results are known as folklore by experts in hyperspaces. In this paper, we deal with a crucial question in this context, which is still unsolved in its full generality: when does the cocompact topology coincide with the topologization of the upper Kuratowski convergence? Such a problem has been already approached in [3,6,7], where some important results are established. We prove here that for a large class of separable metrizable spaces, including the rational line, there is not coincidence. * Corresponding author. E-mail: [email protected]. This article was written while the author was visiting the University of York, by a grant of the Consiglio Nazionale delle Ricerche of Italy. 0166-8641/98/$19.00 0 1998 Published by Elsevier Science. B.V. All rights reserved. PIISO166-8641(97)00096-5 260 C. Costantini, S. Watson / Topology and its Applications 84 (1998) 259-268 1. Definitions and basic results If X is a topological space, we denote by c(X) the set of all closed subsets of X. If M is any subset of X, we put M + = {C E c(X) ) C C: M}. It is easily seen that the collection n, = {(Kc)+ ) I( is a compact subset of X} is closed under finite intersection, and hence is a base for a topology C on c(X), which is called the cocompact topology. For every net (Aj)jE~ of elements of c(X), we put: LQJA~ = {IC E X 1W nbhd of 2: ‘dj E J: gj’ E J, j’ 2 j: V n Ajl # S}. The set Ls~~JA~ is called the upper Kuratowski limit of the net (AJ)jE~; observe that Ls~~JA~ = n,,, Ujlaj A,?. Also, for every net (Aj)jE~ in c(X) and for every A E c(X), we say that (Aj)jE~ converges to A with respect to the upper Kuratowski convergence (in symbols, (Aj)jeJ % A) if LsjE.1A.j C A. Let r be the collection of all subsets C of c(X) such that for every net (Aj)jG~ of elements of C and for every A E c(X), if (A~)~EJ z A then A E C (thus, r is the collection of the subsets of c(X) which are closed with respect to the upper Kuratowski convergence). It is easily shown that r is closed with respect to finite unions and arbitrary intersections, and hence it is the collection of the closed sets for a suitable topology TK+ on c(X)-the upper Kuratowski topology. It is easy to observe that Thus, we always have that C 6 TK+; a topological space X is said to be consonant if TKf = C on c(X), and dissonant otherwise. Let us also remember that if X is a T2 first-countable space, then for every sequence (A71)nEd in c(X) and for every A E c(X), the three relations (JL)~E~ 2 A, (AJnEwTK+ A, and (&h, z A are equivalent. Finally, let us point out that, by [4, Proposition 2.61, a separable metrizable space X is consonant if and only if the cocompact topology on c(X) is sequential. 2. The main result In [6] it is proved that every tech-complete space-hence, in particular, every com- pletely metrizable space-is consonant. On the other hand, Nogura and Shakhmatov [7, Examples 9.1 and 9.31 first proved that there exist met&able dissonant spaces; in the same paper (remark added in proof) it is announced a result by Alleche and Calbrix, prov- ing the existence of a dissonant, hereditarily Baire, separable metrizable space. Finally, Bouziad [3] (and a little bit later Fremlin) showed that the rational line is dissonant. C. Costantini, S. Watson / Topology and its Applications 84 (1998) 259-268 261 We prove here the following fact: Theorem 1. Every (strongly) O-dimensional separable metriz.able space X without iso- lated points. such that every compact subset of it is scattered, is dissonant. Such a result will follow from a series of constructions and lemmas we are going to carry out in this section. Thus, suppose X is a space satisfying the hypotheses above: by the observations of the preceding section, it will suffice to construct a collection C of closed sets of X, with 0 $ C and which is sequentially closed with respect to the upper Kuratowski convergence, such that for every compact subset K of X there is a C E C which is disjoint from K. Before defining the collection C, we need a preliminary lemma. In the following, whenever partitions are concerned, they are always intended not to contain the empty set. Lemma 2. There exists a sequence (Vn)nEw of open partitions of X such that Vo = {X} and: (a) Yn E w: Vn+l < V,; (b) Yx E X: {St(x,Vn) 1 n E w } ts a f un d amental system of open neighbourhoods of (c) ;; E w: VA E V,: card{B f V,,, ) B C A} = 2 (property (c) states that, passing from any partition to the subsequent one, evev element of the first partition is splitted exactly into two parts). Proof. As X is O-dimensional of weight No, there exists a topological embedding j of X into the Cantor set 2” (where 2 = (0, 11). For every cp E 2<w, let hG = {f E zwI fldom(p) = cp}; then J/r, is a closed-and-open subset of 2” and, putting for every n E w, W, = {Ad9 1 cp E 2”, dom(cp) = n}, we have that (W) nEw is a sequence of open partitions of 2“’ with Wa = {2w} and which satisfies conditions (a)-(c) (with W, in the place of V,). For every n E w, put u, = {cp-‘(IY) I W E TV,}; then Ua = {X}, and using the fact that j is a topological embedding it is easily shown that (U 7L) nEw is a sequence of open partitions of X which satisfies conditions (a) and (b) (with 24, in the place of V,). Condition (c) is not necessarily true, but we have a weaker form of it: (c’) Yn E LJ: ‘dA E Z&: (card{B E Z&+1 I B & A} = 1 or 2). Now we will use the fact that X has no isolated point, to modify the partitions U, in such a way that property (c) is satisfied. Put Z4 = UnEw L&, and for every A E 24 put: n.(A) = min{n E w 1card{B E U,, ( B C A} = 2}. Such a minimum exists by properties (a), (c’) and (b), and the fact that X has no isolated point. For every A E U, let Br (A) and B*(A) be the two elements 262 C. Costantini. S. Watson / Topology and its Applications 84 (1998) 259-268 of {B E &(A) 1 B & A}. w e will define by induction a sequence (Vn)nEu, of open partitions of X in the following way: put VO = {X}, and for every n E w put Vn+l = U&V, (&(A), &(A)). 0 ne can prove simultaneously by induction on 71 that this definition is correct (in the sense that every V, is a subset of U, and hence that there exists Bi (A) and &(A) f or every A E V,), that every V, is an open partition of X and that V n+l < V, for every n. Also, we can prove by induction that V, < U, for every n, and hence property (b) for the sequence (Vn)nEw follows from property (b) for the sequence (Z&)nEw. Finally, property (c) is an immediate consequence of the inductive definition. 0 In the following, for the sake of simplicity, we will associate to every z E X and n E w the set V(z, n), which is the unique element of V, containing 2; clearly, V(x, n) = St(z, V,), and hence for every 2 E X the collection { V(z, n) 1n E w} is a fundamental system of neighbourhoods for IC. For every n, m E w and A E V,, define the collection P”(n,A) = {A’ E Vn+m ) A’ C A}; it follows by property (c) that each Pm(n, A) contains exactly 2m sets.
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