Selective sequential pseudocompactness

Abstract

We say that a topological space X is selectively sequentially pseudocompact if for every family {Un:n∈N} of non-empty open subsets of X, one can choose a point xn∈Un for every n∈N in such a way that the sequence {xn:n∈N} has a convergent subsequence. We show that the class of selectively sequentially pseudocompact spaces is closed under arbitrary products and continuous images, contains the class of all dyadic spaces and forms a proper subclass of the class of strongly pseudocompact spaces introduced recently by García-Ferreira and Ortiz-Castillo. We investigate basic properties of this new class and its relations with known compactness properties. We prove that every ω-bounded (= the closure of each countable set is compact) group is selectively sequentially pseudocompact, while compact spaces need not be selectively sequentially pseudocompact. Finally, we construct selectively sequentially pseudocompact group topologies on both the free group and the free Abelian group with continuum-many generators.