Path connectedness, local path connectedness and contractibility of $\mathcal{S}_c(X)$
Volume 160 / 2020
Colloquium Mathematicum 160 (2020), 183-211
MSC: 54A20, 54B20.
DOI: 10.4064/cm7516-1-2019
Published online: 24 January 2020
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, we study the path connectedness and contractibility of $\mathcal {S}_c(X)$. We present necessary conditions on $X$ for the path connectedness of $\mathcal {S}_c(X)$, and also some sufficient conditions. 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 $\mathcal {S}_c(X)$.
