Denumerable cellular families in Hausdorff spaces and towers of Boolean algebras in $\mathbf{ZF}$

Kyriakos Keremedis, Eliza Wajch

Published 2020-01-02Version 1

A denumerable cellular family of a topological space $\mathbf{X}$ is an infinitely countable collection of pairwise disjoint non-empty open sets of $ \mathbf{X}$. It is proved that the following statements are equivalent in $\mathbf{ZF}$: (i) For every infinite set $X,[X]^{<\omega }\mathbf{\ }$has a denumerable subset. (ii) Every infinite $0$-dimensional Hausdorff space admits a denumerable cellular family. It is also proved that (i) implies the following: (iii) Every infinite Hausdorff Baire space has a denumerable cellular family. Among other results, the following theorems are also proved in $\mathbf{ZF}$: (iv) Every countable collection of non-empty subsets of $\mathbb{R}$ has a choice function iff, for every infinite second-countable Hausdorff space $ \mathbf{X}$, it holds that every base of $\mathbf{X}$ contains a denumerable cellular family of $\mathbf{X}$. (v) If every Cantor cube is pseudocompact, then every non-empty countable collection of non-empty finite sets has a choice function. (vi) If all Cantor cubes are countably paracompact, then (i) holds. Moreover, among other forms independent of $\mathbf{ZF}$, a partial Kinna-Wagner selection principle for families expressible as countable unions of finite families of finite sets is introduced. It is proved that if this new selection principle and (i) hold, then every infinite Boolean algebra has a tower and every infinite Hausdorff space has a denumerable cellular family.