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 $K\cap U\ne \emptyset$ 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) $MA{{\omega}_1}$ implies that every point of a compact $T_2$ space of countable spread has a disjoint local $\pi$-base.