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

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 García-Ferreira, Ortiz-Castillo and Rojas-Hernández, 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 ${\mathcal{S}}_c(X)$.