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 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 S c (X) is pathwise connected, in the current paper we study the latter property and the contractibility of S c (X). We present necessary conditions on a space X to obtain the path connectedness of 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 S c (X) in terms of that of X. We prove the contractibility of S c (X) for a class of spaces and, finally, we study the connectedness of Whitney blocks and Whitney levels for S c (X).Question 0.1. What kind of topological properties must X have when the hyperspace S c (X) is pathwise connected? Problem 0.2. Give conditions on X under which S c (X) must be pathwise connected.