Weak pseudocompactness on spaces of continuous functions

A space X is weakly pseudocompact   if it is GδGδ-dense in at least one of its compactifications. X has property  DYDY if for every countable discrete and closed subset N of X  , every function f:N→Yf: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)Cp(X,Y) is weakly pseudocompact where X and Y   are such that Cp(X,Y)Cp(X,Y) is dense in YXYX. We prove: (1) For spaces X and Y such that X   has property DYDY and Y   is first countable weakly pseudocompact and not countably compact, the following conditions are equivalent: (i) Cp(X,Y)Cp(X,Y) is weakly pseudocompact, (ii) Cp(X,Y)Cp(X,Y) is O-pseudocomplete, and (iii) Cp(X,Y)Cp(X,Y) is T-pseudocomplete. (2) For a space X and a compact metrizable topological group G such that X   has property DGDG, the following statements are equivalent: (i) Cp(X,G)Cp(X,G) is pseudocompact, (ii) Cp(X,G)Cp(X,G) is weakly pseudocompact, (iii) Cp(X,G)Cp(X,G) is T  -pseudocomplete, and (iv) Cp(X,G)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.