Denumerable cellular families in \(\mathbf{ZF}\)
Summary: 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.
scientific article; zbMATH DE number 7478506