On first countable, cellular-compact spaces

István Juhász, Lajos Soukup, Zoltán Szentmiklóssy

Published 2019-10-31Version 1

A topological space $X$ is cellular-compact if given any cellular family $\mathcal U$ of 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 two questions of Tkachuk and Wilson we show that (1) if $X$ is a first countable cellular-compact $T_2$ space, then $|X|\le 2^{\omega}$, (2) if $cov(\mathcal M)>{\omega}_1$, then every first countable separable $\pi$-regular cellular-compact space is compact.