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