Selective separability: general facts and behaviour in countable spaces

A space X is called selectively separable if, for any sequence of dense subsets Dn of X, we can choose a finite subset Fn of Dn for each n in such a way that the union of Fn is dense in X; this notion was introduced by Mar- ion Scheepers [Combinatorics of open covers. VI. Selectors for sequences of dense sets, Quaest. Math. 22 (1999), no. 1, 109{130]. Every space of countable $\pi$-weight is selectively separable and if X is selectively separable, then all dense sub- sets of X are separable. We study the general properties of selective separability together with the behavior of this notion in some special classes, such as function spaces or countable spaces. We prove, in particular, that some dense countable subsets of ${0,1}^c$ are selectively separable and some are not. We also show that Cp(X) is selectively separable if and only if it is separable and has countable fan tightness and we give a consistent example of a countable regular maximal space which is not selectively separable.