On first countable, cellular-compact spaces

TL;DR

This paper investigates properties of first countable cellular-compact spaces, establishing conditions under which such spaces are regular, compact, and have bounded cardinality, thus advancing the understanding of their topological structure.

Contribution

It proves that first countable cellular-compact $T_2$ spaces are $T_3$ with bounded cardinality, and explores conditions for their compactness and local base properties.

Findings

First countable cellular-compact $T_2$ spaces are $T_3$ with cardinality at most continuum.

Under certain set-theoretic assumptions, such spaces are compact.

Spaces of countable spread under no $S$-space condition are compact.

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.