The density topology can be not extraresolvable (Q1772959)

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, \textit{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.