  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.
