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

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 $\mathcal{S}_c(X)$.