On Supercomplete Uniform Spaces

Abstract

Abstract.We show that a uniform space iiA" is supercomplete if, and only if, the Ginsburg-Isbell locally fine coreflection of p.X X 'Sß X is equinormal.1. Introduction. J. Isbell calls a uniform space pX supercomplete if the uniform hyperspace H(pX) of closed subsets of X is complete. He proved in [4] that pX is supercomplete if, and only if, X is paracompact and Xp = ®sx, where ^Fx is the fine uniformity of X. (On the other hand, Morita proved in [7] that the uniform hyperspace K(p.X) of all compact subsets of X is complete if and only if p. X is complete.) Tamaño established in 1960 (see [9]) the following famous result: a completely regular space X is paracompact whenever X X ßX is normal. It seems reasonable to ask if it is possible to give a Tamano-like "finitary" characterization of supercomplete uniform spaces by means of their Cech-Stone compactifications. First of all, we will recall some familiar definitions. A quasiuniformity is a filter of coverings ordered by the relation of refinement. If p and v are quasiuniformities, then p/v is the quasiuniformity consisting of all covers {V, D IV'}, where {V¡}¡ G v and for each i, [Wf}j G p. If p is a quasiuniformity, let ju(0) = p. If //0) is defined for all a < ß and ß is a limit ordinal, let p{ß) = U {pia): a < ß). On the other hand, if ß -a + 1, then we define p(ß) = p{a)/p<a). There is an ordinal a such that w(a) = /i<a+" and for which we define Xp = p(a). In case p is a uniformity, it can be shown that [1, Theorem 4.4] Xp is the coarsest locally fine [1] uniformity finer than p. The operator X is called the Ginsburg-Isbell locally fine coreflection. The precompact reflection of p will be denoted by pp. A uniform space is equinormal if every binary open cover of the space is uniform. The symbol °Y | A (resp. p | A) denotes the trace of the family ^ on A (resp. the induced quasiuniformity of A).2. The result. Tamaño proved the following lemma in [9]. Lemma 1. A completely regular space X is paracompact if and only if for each compact K G ßX ~ X there is a locally finite open cover ^ of X such that dpx V (~l K = 0 for all V G % We will extend Lemma 1 to "X-uniform" covers.

On Supercomplete Spaces. III

“…In the context of extending functions defined on certain subspaces of Cartesian products, Husek and Pelant [3] essentially extended Isbell's result by showing that the locally fine coreflection of any product of fine Cech-complete paracompact spaces is fine. Their results include similar results for (C-) scattered paracompact spaces, extending the finite case considered in [2]. However, an interesting problem remains: Is the locally fine coreflection determined by its action on finite subproducts in case the factors are supercomplete or at least separable metrizable?…”

Countable Products of Čech-scattered supercomplete spaces

“…If B were an infinite branch of T K (X), then {r(P ): P ∈ B} would contain an infinite decreasing set of ordinal numbers, which would be impossible. The decomposition trees (for K = C and for a metrizable X) were first explicitly mentioned in [17].…”

“…However, under suitable conditions (see, for example, [13], [14], [16]) it is possible to refine all normal covers of X × Y by covers obtained from combinatorial refinements of product covers, i.e., the equation…”

“…Since uniformly continuous perfect pre-images of supercomplete spaces are again supercomplete ( [16]), one can ask whether the problem of countable products could be reduced to the metrizable case. Unfortunately, there arě C -scattered paracompacta which are not perfect pre-images of metrizable spaces; for example, we can take the 1-point Lindelöfization of an uncountable discrete space.…”

“…A similar result for paracompact spaces was given in [5], extending an earlier result of [28] for scattered paracompact spaces. On the other hand, the earlier result of Telgársky [30] on finite products of C-scattered paracompact spaces was extended to the stronger, combinatorial condition of supercompleteness [21] by the first author in [16]. Following this research line ( [14], [16], [17], [26]), the first author and J. Pelant gave in [19] the result establishing the supercompleteness of countable products of C-scattered supercomplete spaces, which implies the above-mentioned results of [1], [6], [5], and [28].…”