On Properties Characterizing Pseudo- Compact Spaces

Completely regular pseudo-compact spaces have been characterized in several ways. E. Hewitt [6, pp. 68-70] has given one characterization in terms of the Stone-Cech compactification and another in terms of the zero sets of continuous functions. J. Colmez [2; no proofs included] and I. Glicksberg [4] have obtained characterizations by means of a convergence property for sequences of continuous functions (Dini's theorem) and in terms of sequences of closed neighborhoods. A very elegant characterization in terms of a covering property has been obtained by S. Mardesic and P. Papic [8]. This characterization is discussed by K. Iseki and S. Kasahara in [7].1 In this paper we obtain an additional characterization of pseudocompact spaces by means of a convergence property for sequences of continuous functions and characterize completely regular pseudocompact spaces by a covering property. In Theorem 1 we establish the equivalence of many of the topological characterizations of pseudo-compactness (for completely regular spaces) in arbitrary topological spaces. Among the applications of these results is a theorem concerning products of pseudo-compact spaces. We omit strong separation axoms in the definitions of some common terms; e.g., a space X (not necessarily completely regular) is pseudo-compact [6] if every continuous real-valued function on A is bounded, and a space A (not necessarily Hausdorff) is paracompact [9] if every open covering has a locally finite refinement. Definition. A topological space X is lightly compact if every locally finite collection of open sets of X is finite. A topological space is countably compact if every countable open covering has a finite subcovering. We will see (Theorem 1) that a countably compact space is lightly compact. There are examples [6, p. 69] of lightly compact, locally compact, completely regular Tt spaces which are not countably compact.