Abstract
The hyperspace of nontrivial convergent sequences of a metric space X without isolated points will be denoted by Sc(X). This hyperspace is equipped with the Vietoris Topology. It is not hard to prove that Sc([0,1]) and Sc(I) are not homeomorphic, where I are the irrationals. We show that the hyperspaces Sc(R) and Sc([0,1]) are path-wise connected. In a more general context, we show that if X is path-wise connected space, then Sc(X) is connected. But Sc(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 Sc(X) is meager. We list several open questions concerning this hyperspace.
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