Almost Disjointness Principles and $Q$-Space Cardinals

Abstract

Banakh and Bazylevych introduced separation-axiom variants ${\mathfrak{q}}_i$, for $i=1,2,2\frac{1}{2}$, of the cardinal $\mathfrak{q}$, together with a cardinal $\mathfrak{adp}$ lying between $\mathfrak{dp}$ and $\mathfrak{ap}$. They asked whether $\mathfrak{adp}$ coincides with either of these two cardinals. We prove in ZFC that $\mathfrak{adp}=\mathfrak{dp}$. We define a dual variant ${\mathfrak{adp}}_2$ and show that ${\mathfrak{adp}}_2=\mathfrak{ap}$. We further study the relation between $\mathfrak{ap}$ and the weakened $Q$-space cardinals. We introduce a tree analogue $\mathfrak{at}$ of $\mathfrak{ap}$ and prove $\mathfrak{at}={\mathfrak{q}}_2$, concluding that $\mathfrak{ap}\le {\mathfrak{q}}_2$, which answers a question of Banakh and Bazylevych. Assuming the Generalized Continuum Hypothesis, we construct ccc forcing extensions with $\mathfrak{ap}={\omega }_1<\mathfrak{at}=\mathfrak{c}$, so $\mathfrak{ap}<{\mathfrak{q}}_2$ is consistent with ZFC.

28 pages. Revised version: proves $\mathfrak{at}={\mathfrak{q}}_2$ and answers Problem 11(1) of Banakh and Bazylevych; adds results on $G_{\delta }$-diagonals and further questions