The Baire property on the hyperspace of nontrivial convergent sequences

Abstract

In this paper, we shall consider the hyperspace of all nontrivial convergent sequences Sc(X) of a Fréchet-Urysohn nondiscrete space X, which is equipped with the Vietoris topology. We study the spaces X for which Sc(X) is Baire: this kind of spaces have a dense subset of isolated points (see [12]). We characterize the spaces X for which Sc(X) is Baire. This characterization uses a topological game inspired by the Banach-Mazur game. As a consequence, we obtain that if X is completely metrizable and has a dense subset of isolated points, then Sc(X) is Baire. Our last main result shows that Sc(X) is pseudocompact iff X is homeomorphic to the one-point compactification of a discrete space of uncountable size. This last assertion provides a characterization of the one-point compactification of a discrete space of uncountable size.