Selective separability and $q^+$ on maximal spaces

Abstract

Given a hereditarily meager ideal $\mathcal{I}$ on a countable set $X$ we use Martin's axiom for countable posets to produce a zero-dimensional maximal topology $\tau^\mathcal{I}$ on $X$ such that $\tau^\mathcal{I}\cap \mathcal{I}=\{\emptyset\}$ and, moreover, if $\mathcal{I}$ is $p^+$ then $\tau^\mathcal{I}$ is selectively separable (SS) and if $\mathcal{I}$ is $q^+$, so is $\tau^\mathcal{I}$. In particular, we obtain regular maximal spaces satisfying all boolean combinations of the properties SS and $q^+$.