Categorical properties on the hyperspace of nontrivial convergent sequences

Abstract

In this paper, we shall study categorial properties of the hyperspace of all nontrivial convergent sequences $\mathcal{S}_c(X)$ of a Fre\'ech-Urysohn space $X$, this hyperspace is equipped with the Vietoris topology. We mainly prove that $\mathcal{S}_c(X)$ is meager whenever $X$ is a crowded space, as a corollary we obtain that if $\mathcal{S}_c(X)$ is Baire, the $X$ has a dense subset of isolated points. As an interesting example $\mathcal{S}_c(\omega_1)$ has the Baire property, where $\omega_1$ carries the order topology (this answers a question from \cite{sal-yas}). We can give more examples like this one by proving that the Alexandroff duplicated $\mathcal{A}(Z)$ of a space $Z$ satisfies that $\mathcal{S}_c(\mathcal{A}(Z))$ has the Baire property, whenever $Z$ is a $\Sigma$-product of completely metrizable spaces and $Z$ is crowded. Also we show that if $\mathcal{S}_c(X)$ is pseudocompact, then $X$ has a relatively countably compact dense subset of isolated points, every finite power of $X$ is pseudocompact, and every $G_\delta$-point in $X$ must be isolated. We also establish several topological properties of the hyperspace of nontrivial convergent sequences of countable Fre\'ech-Urysohn spaces with only one non-isolated point.