Selective separability on spaces with an analytic topology

TL;DR

This paper investigates selective separability properties in countable analytic spaces, establishing equivalences with Ramsey properties, $G_\delta$ conditions, and $F_\sigma$-bases, and explores various examples.

Contribution

It introduces new characterizations of $SS$ and $SS^+$ in analytic spaces, linking them to Ramsey properties and topological bases.

Findings

$SS$ follows from Ramsey-type properties.

$SS^+$ is equivalent to the collection of dense sets being a $G_\delta$.

Examples of analytic spaces illustrating these properties.

Abstract

We study two form of selective selective separability, $SS$ and $SS^+$, on countable spaces with an analytic topology. We show several Ramsey type properties which imply $SS$. For analytic spaces $X$, $SS^+$ 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.