On Properties Characterizing Pseudo-Compact Spaces

Pseudocompact Spaces

“…Terasaka's example (see [8, p. 135]) shows that if X is a countably compact completely regular space, then XxX need not be pseudocompact. Comfort's example in [4] shows that if {X(h) \ neN} are completely regular spaces, then n {X(n) \ne N} is not necessarily pseudocompact even if F] {X(n) | n e B) is pseudocompact for every finite (nonempty) subset B of N. On the other hand, Glicksberg [9] and Bagley, Connell, and McKnight [1] have proved that under certain supplementary hypotheses the product of a collection of pseudocompact completely regular spaces is pseudocompact.…”

“…Definition 4.2. A space A-is said to be feebly compact [13], or lightly compact [1], provided that Zhas property B(l).…”

“…By definition, a pseudocompact space is a topological space on which every continuous real valued function is bounded. Pseudocompact spaces which are completely regular have often been studied [1], [4], [5], [7]- [10], [12], and [15], and some results have been obtained that apply to noncompletely regular pseudocompact spaces [1], [10], and [13]. In general, however, much more is known about completely regular pseudocompact spaces than is known about arbitrary pseudocompact spaces.…”

“…We shall denote the set of continuous mappings of a space (X, V) into a space (Y, iT) by C((X, Y~), (Y, W)) or C(X, Y) if no confusion is possible. C(X, f") or C(X) will denote the set of bounded functions in C((X, V), R), and L(X, *T) or L(X) will denote the set of all functions in C(X, "T) which map (X, *T) into [0,1]. Given afunction/e C((X, -V), R), we shall denote the zero set/_1(0) by Z(f) and the cozero set X-Z(f) by C(f).…”

“…In [1] and [14] examples were constructed in order to show that on a Fi-space B(l) is not a necessary condition for pseudocompactness. This result can be sharpened.…”