Some remarks on extraresolvable spaces
A space \(X\) is said to be extraresolvable if there is a family \(\mathcal D\) of dense subsets of \(X\) such that \(|\mathcal D|>\Delta(X)\) and the intersection of any two distinct members of \(\mathcal D\) is nowhere dense, where \(\Delta(X)=\min\{|U|:U\) is a nonempty open subset of \(X\}\). A space \(X\) is said to be strongly extraresolvable if there is a family \(\mathcal D\) of dense subsets of \(X\) such that \(|\mathcal D|>\Delta(X)\) and the intersection of any two distinct members of \(\mathcal D\) has size \(<\text{nwd}(X)\), where \(\text{nwd}(X)=\min\{|A|:A\subseteq X\) and \(\text{int}_X(\text{cl}_XA)\neq \emptyset\}\). The authors give an example of a countable extraresolvable space that is not strongly extraresolvable. They find some sufficient conditions and some necessary conditions for spaces of the form \(X\times\alpha\) to be strongly extraresolvable. In particular, they show that \(\mathbb Q\times\omega_1\) is strongly extraresolvable and if \(X\) is strongly extraresolvable and \(\text{nwd}(X)=\Delta(X)=\omega\), then \(X\times\omega\) is strongly extraresolvable.
scientific article; zbMATH DE number 1787556