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 ${\tau }^{\mathcal{I}}\cap \mathcal{I}=\{\varnothing \}$ 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 $q^{+}$.

17 pages