On cellular-compact spaces

Abstract

As it was introduced by Tkachuk and Wilson in [7], a topological space X is cellular-compact if for any cellular, i.e. disjoint, family $\mathcal{U}$ of non-empty open subsets of X there is a compact subspace $K \subset X$ such that $K \cap U \ne \emptyset$ for each $U \in \mathcal{U}$.

In this note we answer several questions raised in [7] by showing that

• (1) any first countable cellular-compact T₂-space is T₃, and so its cardinality is at most $\mathfrak{c} = 2^{\omega}$;

• (2) cov$(\mathcal{M}) > {\omega}_1$ implies that every first countable and separable cellular-compactT₂-space is compact;

• (3) if there is no S-space then any cellular-compact T₃-space of countable spread is compact;

• (4) $MA_{\omega{1}}$ implies that every point of a compact T₂-space of countable spread has a disjoint local $\pi$-base.