On a variation of selective separability: S-separability

A space $X$ is M-separable (selectively separable) (Scheepers, 1999; Bella et al., 2009) if for every sequence $(Y_n)$ of dense subspaces of $X$ there exists a sequence $(F_n)$ such that for each $n$ $F_n$ is a finite subset of $Y_n$ and ${\cup }_{n\in \mathbb{N}}{F}_n$ is dense in $X$. In this paper, we introduce and study a strengthening of M-separability situated between H- and M-separability, which we call S-separability: for every sequence $(Y_n)$ of dense subspaces of $X$ there exists a sequence $(F_n)$ such that for each $n$ $F_n$ is a finite subset of $Y_n$ and for each finite family $\mathcal{F}$ of nonempty open sets of $X$ some $n$ satisfies $U\cap F_n\mathrm{\ne }\mathrm{\varnothing }$ for all $U\in \mathcal{F}$.