Denumerable cellular families in $\mathbf{ZF}$

Abstract

A denumerable cellular family of a topological space $\mathbf{X}$ is a countably infinite collection of pairwise disjoint non-empty open sets of $\mathbf{X}$. $\mathbf{IQDI}$ is the sentence: For every infinite set $X$, the set of all finite subsets of $X$ has a countably infinite subset. Among other results, the following are proved in $\mathbf{ZF}$: (i) $\mathbf{IQDI}$ iff every infinite 0-dimensional Hausdorff space admits a denumerable cellular family; (ii) $\mathbf{IQDI}$ implies that every infinite Hausdorff Baire space has a denumerable cellular family; (iii) if every metrizable Cantor cube is pseudocompact, then every non-empty countable collection of non-empty finite sets has a choice function; (iv) all metrizable Cantor cubes are paracompact; (v) the conjunction of $\mathbf{IQDI}$ with a new modification of the Kinna–Wagner selection principle for families of finite sets implies that every infinite Boolean algebra has a tower and every infinite Hausdorff space has a denumerable cellular family.