Some Cardinality Properties of a Hyperspace with the Locally Finite Topology

Estimates of various cardinal functions defined on a hyperspace equipped with the locally finite topology are given. The locally finite topology on a hyperspace is the natural generalization of the well-known Vietoris or finite topology. Recently some authors (see [1 and 9]) have investigated its properties finding interesting relationships with the Hausdorff metric topology, when the base space is metrizable, and the uniform topology, when the base space is normal. In this paper we are concerned with some cardinality properties of a hyperspace with the locally finite topology. Specifically we provide estimates of various cardinal functions defined on it. Some results of the same kind relative to the finite case can be found in [3, 6, and 8]. For notation and terminology we refer to [5]. Each cardinal number is assumed to be an initial ordinal. As usual, given two cardinal numbers A, d, d<x is defined as sup{/tfM : p < A}. |S| denotes the cardinality of the set S and ¿P(S) its powerset. [S]k ([S]<k or [S]-fc) is the collection of all subsets of S having cardinality d (less than /, or less or equal to 4). Throughout the paper A always denotes a topological Ti space. c(A), d(A), ?r(A), w(X), L(X), x(A) and 7rx(A) denote, respectively, the cellularity, and density, the 7r-weight, the weight, the Lindelöf number, the character and the ttcharacter of A. x(A, A) denotes the character of the subset A of A. For definitions and more details concerning these cardinal functions we refer to the Juhász's book [7]. All other cardinal functions used here will be explicitly defined. DEFINITION 1. The weak covering number of X, denoted by wc(A), is the smallest infinite cardinal number d such that every open cover of A has a subfamily of cardinality at most d whose union is dense in A. DEFINITION 2. The pseudocompactness number of A, denoted by p(A), is the smallest infinite cardinal number d such that every locally finite family of open subsets of X has cardinality at most d'. To formulate some of our results in a little stronger way than using exponentiation by p(A) or L(X), we need also the following (see [7, 1.22]): DEFINITION 2'. p(A) is the smallest infinite cardinal number / such that in A there exists no locally finite family of open sets of cardinality d'. Clearly A is pseudocompact if and only if it is Tychonov and p(A) = No [5, p. 263]. Received by the editors June 9, 1985 and, in revised form, September 10, 1985. 1980 Mathematics Subject Classification (1985 Revision). Primary 54A25, 54B20. ©1988 American Mathematical Society 0002-9939/88 $1.00 + $.25 per page