The Baire property on the hyperspace of nontrivial convergent sequences (Q820643)

A space in this paper is assumed to be a Fréchet-Urysohn nondiscrete completely regular Hausdorff space. An infinite countable subset \(S\) of a space \(X\) is called a nontrivial convergent sequence if \(S\) has a unique non-isolated point \(x_S\) such that \(S \setminus U\) is finite for each neighborhood \(U\) of \(x_S\). Let \(\mathcal{S}_c(X)\) denote the hyperspace consisting of all nontrivial convergent sequences of a space \(X\) with the Vietoris topology. In this paper, the authors give a characterization of \(\mathcal{S}_c(X)\) being a Baire space and a characterization of \(\mathcal{S}_c(X)\) being pseudocompact, which answers questions posed in [the authors, Topol. Proc. 52, 265--279 (2018; Zbl 1393.54011)]. In fact, the authors define a topological game \(GS(X,\beta,\alpha)\) of two players \(\alpha\) and \(\beta\) like the Banach-Mazur game, and prove that \(\mathcal{S}_c(X)\) is a Baire space if and only if \(X\) has a dense set of isolated points and \(X\) does not admit a winning strategy for player \(\beta\) in the game \(GS(X,\beta,\alpha)\). As corollaries of this theorem, the following are obtained: if a space \(X\) is completely metrizable and has a dense subset of isolated points, then \(\mathcal{S}_c(X)\) is a Baire space; if \(X\) is a scattered metric space, then \(\mathcal{S}_c(X)\) is a Baire space. It is also proved that \(\mathcal{S}_c(X)\) is pseudocompact if and only if \(X\) is homeomorphic to the one-point compactification of some uncountable discrete space.