Weak pseudocompactness on spaces of continuous functions

Abstract

A space X is weakly pseudocompact if it is Gδ-dense in at least one of its compactifications. X has propertyDY if for every countable discrete and closed subset N of X, every function f:N→Y can be continuously extended to a function over all of X. O-pseudocompleteness is the pseudocompleteness property defined by J.C. Oxtoby [17], and T-pseudocompleteness is the pseudocompleteness property defined by A.R. Todd [24].In this paper we analyze when a space of continuous functions Cp(X,Y) is weakly pseudocompact where X and Y are such that Cp(X,Y) is dense in YX. We prove: (1) For spaces X and Y such that X has property DY and Y is first countable weakly pseudocompact and not countably compact, the following conditions are equivalent: (i) Cp(X,Y) is weakly pseudocompact, (ii) Cp(X,Y) is O-pseudocomplete, and (iii) Cp(X,Y) is T-pseudocomplete. (2) For a space X and a compact metrizable topological group G such that X has property DG, the following statements are equivalent: (i) Cp(X,G) is pseudocompact, (ii) Cp(X,G) is weakly pseudocompact, (iii) Cp(X,G) is T-pseudocomplete, and (iv) Cp(X,G) is O-pseudocomplete. (3) For every space X, Cp⁎(X) and Cp⁎(X,Z) are not weakly pseudocompact. Throughout this study we also consider several completeness properties defined by topological games such as the Banach–Mazur game and the Choquet game.