The Baire category of the hyperspace of nontrivial convergent sequences

TL;DR

This paper investigates the topological complexity of the space of nontrivial convergent sequences in a regular space, showing it is of the first category if the space has no isolated points, answering a prior open question.

Contribution

It proves that the hyperspace of nontrivial convergent sequences is of the first category in spaces without isolated points, resolving an open problem in the field.

Findings

$S_c(X)$ is of the first category if $X$ has no isolated points

Provides a negative answer to the posed question by Garcia-Ferreira and Ortiz-Castillo

Enhances understanding of the topological structure of hyperspaces of convergent sequences

Abstract

Assume that $X$ is a regular space. We study topological properties of the family $S_c(X)$ of nontrivial convergent sequences in $X$ equipped with the Vietoris topology. We show that if $X$ has no isolated points, then $S_c(X)$ is a space of the first category which answers the question posed by S. Garcia-Ferreira and Y.F. Ortiz-Castillo.