The Baire property on the hyperspace of nontrivial convergent sequences

Abstract

In this paper, we shall consider the hyperspace of all nontrivial convergent sequences S c ( X ) of a Fréchet-Urysohn nondiscrete space X, which is equipped with the Vietoris topology. We study the spaces X for which S c ( X ) is Baire: this kind of spaces have a dense subset of isolated points (see [12]). We characterize the spaces X for which S c ( 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 S c ( X ) is Baire. Our last main result shows that S c ( 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.