The hyperspace of convergent sequences

摘要

The hyperspace of nontrivial convergent sequences of a metric space X without isolated points will be denoted by S-c(X). This hyperspace is equipped with the Vietoris Topology. It is not hard to prove that S-c([0,1]) and S-c(I) are not homeomorphic, where I are the irrationals. We show that the hyperspaces S-c(R) and S-c([0,1]) are path-wise connected. In a more general context, we show that if X is path-wise connected space, then S-c(X) is connected. But S-c(X) is not necessarily path-wise connected even when X is the Warsaw circle. These make interesting to study the connectedness of the hyperspace of nontrivial convergent sequences in the realm of continua. Also, we prove that if X is a second countable space, then S-c(X) is meager. We list several open questions concerning this hyperspace. (C) 2015 Elsevier B.V. All rights reserved.