Denumerable cellular families in \(\mathbf{ZF}\)
type: Item
instance of: scholarly article
title: Denumerable cellular families in \(\mathbf{ZF}\)
DOI: 10.4171/PM/2070
author: Eliza Wajch
author: Kyriakos Keremedis
published in: Portugaliae Mathematica. Nova Série
publication date: 2022-02-22T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 03E25
Mathematics Subject Classification ID: 03E35
Mathematics Subject Classification ID: 54A35
Mathematics Subject Classification ID: 54D20
Mathematics Subject Classification ID: 54D70
Mathematics Subject Classification ID: 54E52
zbMATH DE Number: 7478506
zbMATH Keywords: weak forms of the axiom of choice
zbMATH Keywords: Dedekind-finite set
zbMATH Keywords: 0-dimensional Hausdorff space
zbMATH Keywords: Cantor cube
zbMATH Keywords: denumerable cellular family
zbMATH Keywords: Boolean algebra
MaRDI profile type: Publication
cites work: Weak choice principles
cites work: Hausdorff gaps and limits
cites work: On Stone’s theorem and the Axiom of Choice
cites work: Q5185667
cites work: Q4487249
cites work: Non-constructive properties of the real numbers
cites work: Q4210469
cites work: The axiom of choice
cites work: Non-discrete metrics in and some notions of finiteness
cites work: On lightly and countably compact spaces in ZF
cites work: Cellularity of infinite Hausdorff spaces in \textbf{ZF}
cites work: On densely complete metric spaces and extensions of uniformly continuous functions in $\mathbf{ZF}$
cites work: Q3994998
cites work: Q3586892
cites work: Counterexamples in topology
cites work: On the Set-Theoretic Strength of Countable Compactness of the Tychonoff Product 2R
cites work: Infinite Hausdorff spaces may lack cellular families or discrete subsets of cardinality \(\aleph_0\)
cites work: The structure of amorphous sets
cites work: Quasi-metrizability of products in ZF and equivalences of CUT(fin)
description: scientific article; zbMATH DE number 7478506