Connectedness like properties on the hyperspace of convergent sequences

TL;DR

This paper explores how connectedness properties of a space are reflected in the hyperspace of its nontrivial convergent sequences, establishing equivalences and examples related to path-wise connectedness.

Contribution

It generalizes previous results by linking the connectedness properties of a space with those of its hyperspace of convergent sequences, answering open questions.

Findings

Connectedness of X implies connectedness of its hyperspace

Local connectedness of X is equivalent to that of the hyperspace

The hyperspace of the Warsaw circle has continuum many path-wise connected components

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.