Selectively pseudocompact spaces

Abstract

A novel selection principle was introduced by Dorantes-Aldama and Shakhmatov: a topological space $X$ is termed {\em selectively pseudocompact} if for any sequence $(U_n:n\in \omega )$ of pairwise disjoint non-empty open sets of $X$, one can choose points $x_n\in U_n$ such that the sequence $(x_n:n\in \omega )$ has an accumulation point. In this paper, we explore various versions of this principle when we permit the selection of finite, scattered, or nowhere dense sets instead of just singletons. We develop a method to prove that the aforementioned versions of selective pseudocompactness are indeed distinct from one another.

10 pages