JournalspmVol. 78, No. 3/4pp. 281–321

Denumerable cellular families in ZF\mathbf{ZF}

  • Kyriakos Keremedis

    University of the Aegean, Karlovassi, Greece
  • Eliza Wajch

    Siedlce University of Natural Sciences and Humanities, Poland
Denumerable cellular families in $\mathbf{ZF}$ cover

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Abstract

A denumerable cellular family of a topological space X\mathbf{X} is a countably infinite collection of pairwise disjoint non-empty open sets of X\mathbf{X}. IQDI\mathbf{IQDI} is the sentence: For every infinite set XX, the set of all finite subsets of XX has a countably infinite subset. Among other results, the following are proved in ZF\mathbf{ZF}: (i) IQDI\mathbf{IQDI} iff every infinite 0-dimensional Hausdorff space admits a denumerable cellular family; (ii) IQDI\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 IQDI\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.

Cite this article

Kyriakos Keremedis, Eliza Wajch, Denumerable cellular families in ZF\mathbf{ZF}. Port. Math. 78 (2021), no. 3, pp. 281–321

DOI 10.4171/PM/2070