On \(d\)- and \(D\)-separability (Q714749)

A space is \textit{\(d\)-separable} if it has a dense \(\sigma\)-discrete subset. Say that a space \(X\) is \textit{\(D\)-separable} if for any sequence \(\{E_n:n\in\omega\}\) of dense subsets of \(X\), it is possible to choose a discrete set \(D_n\subset E_n\) for each \(n\in\omega\) in such a way that the set \(\bigcup_{n\in\omega}D_n\) is dense in \(X\). Let \(\mathcal G\) be a game played on a space \(X\) as follows: at the \(n\)-th move the player \(I\) chooses a dense set \(E_n\) and the player II responds by taking a discrete set \(D_n\subset E_n\). After the moves \((E_n,D_n)\) are made for all \(n\in\omega\), the player II wins if the set \(\bigcup_{n\in\omega}D_n\) is dense in \(X\); otherwise I is the winner. Call \(X\) a \textit{\(D^+\)-separable space} if the player II has a winning strategy on \(X\). The paper is devoted to study the relationships between \(d\)-separability, \(D\)-separability and \(D^+\)-separability. It is proved, among other things, that a monotonically normal space is \(d\)-separable if and only if it is \(D^+\)-separable. Besides, if \(\pi w(X)<\text{cov}({\mathcal M})\) and every dense subspace of \(X\) is \(d\)-separable, then \(X\) is \(D\)-separable. Here \(\text{cov}({\mathcal M})\) is the least cardinality of a cover of the real line by meager subsets.