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rapid ultrafilter, Q-point, semiselective, rapid point, weakly k-rapid point, countable strong fan tightness, Id-fan tightness, countable fan tightness. 1. Preliminaries.

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.

We dedicate this paper to Prof. Ofelia Alas on her 70th birthday. MSC:primary 54H11, 54B05 secondary 54E99A space X is called strongly pseudocompact if for each sequence (U n ) n∈N of pairwise disjoint nonempty open subsets of X there is a sequence (x n ) n∈N of points in X such that cl X ({x n : n ∈ N}) \ n∈N U n = ∅ and x n ∈ U n , for each n ∈ N. It is evident that every countably compact space is strongly pseudocompact and every strongly pseudocompact space is pseudocompact. In this paper, we construct a pseudocompact group that is not strongly pseudocompact answering two questions posed in [13].