Proper forcing axiom and selective separability
Selectively separable (or SS) spaces are becoming popular nowadays; perhaps less popular are SS\(^+\)-spaces which are those in which player II has a winning strategy for the following natural game: player I picks a dense set \(D_n\); player II picks a finite set \(E_n\subseteq D_n\). Player II wins if \(\bigcup_{n\in\omega}E_n\) is dense.NEWLINENEWLINEIn the paper under review the authors address a question by Gruenhage asking if player II would always have a Markov strategy in each SS\(^+\)-space; that is, a strategy that only depends on the present and the other player's previous move. They solve the problem for countable spaces showing that in such cases the strategy can be chosen to be Markov.NEWLINENEWLINEThey introduce the notion of being \textit{compact-like} for a collection of open sets in a topological space and they use it to show that the union of SS\(^+\)-spaces may fail to be an SS\(^+\)-space and that the property of being a Markov SS-space is finitely productive; this has the consequence that the product of two countable SS\(^+\)-spaces is again of the same kind. A nice result in the paper is that PFA implies that the product of two countable Fréchet spaces is SS. The same authors proved in a previous paper that MA\(_{\;ctble}\) may be used to obtain countable SS-spaces whose product is not SS.NEWLINENEWLINEThe paper closes with some open problems; for instance, the authors restate Gruenhage's question which remains open in the general case as well as the consistency of the fact that the product of two separable Fréchet spaces is SS. Another problem is whether there is a ZFC example of a countable space which is Fréchet but not SS\(^+\).
scientific article; zbMATH DE number 6023696
scientific article