Connectedness like properties on the hyperspace of convergent sequences

Abstract

This paper is a continuation of the work done in \cite{sal-yas} and \cite{may-pat-rob}. We deal with the Vietoris hyperspace of all nontrivial convergent sequences ${\mathcal{S}}_c(X)$ of a space $X$. We answer some questions in \cite{sal-yas} and generalize several results in \cite{may-pat-rob}. We prove that: The connectedness of $X$ implies the connectedness of ${\mathcal{S}}_c(X)$; the local connectedness of $X$ is equivalent to the local connectedness of ${\mathcal{S}}_c(X)$; and the path-wise connectedness of ${\mathcal{S}}_c(X)$ implies the path-wise connectedness of $X$. We also show that the space of nontrivial convergent sequences on the Warsaw circle has $\mathfrak{c}$-many path-wise connected components, and provide a dendroid with the same property.

Cited by in corpus (1)

• Categorical properties on the hyperspace of nontrivial convergent sequences