Completeness type properties on Cp(X,Y) spaces

Abstract

In this paper we improve several results presented in [7] and in [9] related to the characterization of several kinds of pseudocompleteness and compactness properties in spaces of continuous functions of the form Cp(X,Y). In particular, we prove that for every Tychonoff space X and every separable metrizable topological group G for which Cp(X,G) is dense in GX, Cp(X,G) is weakly α-favorable if and only if X is uG-discrete. This result helps us to obtain two generalizations of a theorem due to V.V. Tkachuk in [24]:Theorem I5.5Let G be a separable completely metrizable topological group and X a set. If H is a dense subgroup ofGXand H is homeomorphic toGYfor some set Y, thenH=GX.A particular and interesting case is when X is a Tychonoff space and H=Cp(X,G) is dense in GX.Theorem II6.9Let G be a realcompact Čech-complete weakly α-favorable topological group with countable pseudocharacter and let X be regularC<ωG-discrete. Then,Cp(X,G)≅Gκif and only if X is a discrete space of cardinality κ.Besides, we obtain several applications to weakly pseudocompact spaces.