Virtual Special Issue - Proceedings of the First Pan-Pacific International Conference on Topology and Applications (PPICTA)Connectedness like properties on the hyperspace of convergent sequences

This paper is a continuation of the work done in [3] and [7]. We deal with the Vietoris hyperspace of all nontrivial convergent sequences Sc(X) of a space X. We answer some questions in [3] and generalize several results in [7]. We prove that: The connectedness of X implies the connectedness of Sc(X); the local connectedness of X is equivalent to the local connectedness of Sc(X); and the path-wise connectedness of Sc(X) implies the path-wise connectedness of X. We also show that the hyperspace of nontrivial convergent sequences on the Warsaw circle has c-many path-wise connected components, and provide a dendroid with the same property.