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A denumerable cellular family of a topological space $\mathbf{X}$ is aninfinitely 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 denumerablesubset.  (ii) Every infinite $0$-dimensional Hausdorff space admits a denumerablecellular 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 achoice function iff, for every infinite second-countable Hausdorff space $\mathbf{X}$, it holds that every base of $\mathbf{X}$ contains a denumerablecellular family of $\mathbf{X}$.  (v) If every Cantor cube is pseudocompact, then every non-empty countablecollection 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 partialKinna-Wagner selection principle for families expressible as countable unionsof finite families of finite sets is introduced. It is proved that if this newselection principle and (i) hold, then every infinite Boolean algebra has atower and every infinite Hausdorff space has a denumerable cellular family.

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