On first countable, cellular-compact spaces

Abstract

As it was introduced by Tkachuk and Wilson, a topological space $X$ is cellular-compact if given 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 $K\cap U\ne \varnothing$ for each $U\in \mathcal{U}$. Answering several questions raised by Tkachuk and Wilson we show that (1) any first countable cellular-compact $T_2$ space is $T_3$, 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-compact $T_2$ space is compact; (3 if there is no $S$-space then any cellular-compact $T_3$ space of countable spread is compact; (4) $M{A}_{{\omega }_1}$ implies that every point of a compact $T_2$ space of countable spread has a disjoint local $\pi$-base.

new, revised version, 6 pages