# Categorical properties on the hyperspace of nontrivial convergent
sequences

Research paper by **S. Garcia-Ferreira, R. Rojas-Hernandez, Y. F. Ortiz-Castillo**

Indexed on: **24 Nov '16**Published on: **24 Nov '16**Published in: **arXiv - Mathematics - General Topology**

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#### 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.