Selective separability on spaces with an analytic topology

Abstract

We study two form of selective selective separability, $SS$ and $S{S}^{+}$, on countable spaces with an analytic topology. We show several Ramsey type properties which imply $SS$. For analytic spaces $X$, $S{S}^{+}$ is equivalent to have that the collection of dense sets is a $G_{\delta }$ subset of $2^X$, and also equivalent to the existence of a weak base which is an $F_{\sigma }$-subset of $2^X$. We study several examples of analytic spaces.