All non-𝑃-points are the limits of nontrivial sequences in supercompact spaces

Abstract

A Hausdorff topological space is called supercompact if there exists a subbase such that every cover consisting of this subbase has a subcover consisting of two elements. In this paper, we prove that every non-P-point in any continuous image of a supercompact space is the limit of a nontrivial sequence. We also prove that every non-P-point in a closed G δ G_{\delta } -subspace of a supercompact space is a cluster point of a subset with cardinal number ≤ c . \leq c. But we do not know whether this statement holds when replacing c c by the countable cardinal number. As an application, we prove in ZFC that there exists a countable stratifiable space which has no supercompact compactification.