The density topology can be not extraresolvable. (English) Zbl 1083.28002

Natkaniec, Tomasz

Real Anal. Exch. 30(2004-2005), No. 1, 393-396 (2005).

A topological space \(X\) is called extraresolvable if there exists a family \(\mathcal D\) of dense subsets of \(X\) such that \(D\cap D'\) is nowhere dense for any distinct \(D,D'\in\mathcal D\) and \(| \mathcal D| >\Delta(X)\) where \(\Delta(X)= \min\{| U| :U\) is a nonempty open subset of \(X\}\) is the dispersion character of \(X\). Assuming Martin’s Axiom, A. Bella [Atti Sem.Mat.Fis.Univ.Modena 48, 495–498 (2000; Zbl 1013.54001)] has proved that the real line with the Lebesgue density topology \(\mathcal T_d\) is extraresolvable and asked whether this can be proved in ZFC. In the paper under review the author answers this question in negative proving that if \(\mathfrak c=\omega_2\), \(2^{\omega_1}=\omega_2\), and the cofinality of the ideal of Lebesgue measure zero sets is \(\omega_1\), then \(\mathcal T_d\) is not extraresolvable. Let us remark that all these hypotheses are consequences of the Covering Property Axiom CPA introduced by K. Ciesielski and J. Pawlikowski.

Reviewer: Miroslav Repický (Košice)

MSC:

28A05	Classes of sets (Borel fields, \(\sigma\)-rings, etc.), measurable sets, Suslin sets, analytic sets
03E35	Consistency and independence results
54A25	Cardinality properties (cardinal functions and inequalities, discrete subsets)
26A03	Foundations: limits and generalizations, elementary topology of the line

Keywords:

density topology; extraresolvable space; Covering Property Axiom; CPA

Citations:

Zbl 1013.54001