Categorical properties on the hyperspace of nontrivial convergent sequences

2016/11/24 by Garcia-Ferreira, S., Rojas-Hernandez, R., Ortiz-Castillo, Y. F.
#FOS: Mathematics #General Topology (math.GN)

paper · doi:10.48550/arxiv.1611.08289

Abstract

In this paper, we shall study categorial properties of the hyperspace of all nontrivial convergent sequences S_c(X) of a Freéch-Urysohn space X, this hyperspace is equipped with the Vietoris topology. We mainly prove that S_c(X) is meager whenever X is a crowded space, as a corollary we obtain that if S_c(X) is Baire, the X has a dense subset of isolated points. As an interesting example S_c(ω₁) has the Baire property, where ω₁ carries the order topology (this answers a question from \citesal-yas). We can give more examples like this one by proving that the Alexandroff duplicated A(Z) of a space Z satisfies that S_c(A(Z)) has the Baire property, whenever Z is a Σ-product of completely metrizable spaces and Z is crowded. Also we show that if 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_δ-point in X must be isolated. We also establish several topological properties of the hyperspace of nontrivial convergent sequences of countable Freéch-Urysohn spaces with only one non-isolated point.

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