Path connectedness, local path connectedness and contractibility of $\mathcal{S}_c(X)$

TL;DR

This paper investigates the conditions under which the hyperspace of nontrivial convergent sequences in a Hausdorff space is path connected or contractible, providing new insights into its topological structure.

Contribution

It establishes necessary and sufficient conditions for path connectedness and local path connectedness of (X), and characterizes its contractibility for certain classes of spaces.

Findings

Necessary conditions for path connectedness of (X)

Sufficient conditions for path connectedness of (X)

Contractibility of (X) for specific spaces

Abstract

The hyperspace of all nontrivial convergent sequences in a Hausdorff space $X$ is denoted by $\mathcal{S}_c(X)$. This hyperspace is endowed with the Vietoris topology. In connection with a question and a problem by Garc\'ia-Ferreira, Ortiz-Castillo and Rojas-Hern\'andez, concerning conditions under which $\mathcal S_c(X)$ is pathwise connected, in the current paper we study the latter property and the contractibility of $\mathcal{S}_c(X)$. We present necessary conditions on a space $X$ to obtain the path connectedness of $\mathcal{S}_c(X)$. We also provide some sufficient conditions on a space $X$ to obtain such path connectedness. Further, we characterize the local path connectedness of $\mathcal{S}_c(X)$ in terms of that of $X$. We prove the contractibility of $\mathcal{S}_c(X)$ for a class of spaces and, finally, we study the connectedness of Whitney blocks and Whitney levels for $\mathcal{S}_c(X)$.