Almost Disjointness Principles and $Q$-Space Cardinals

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 $\mathfrak{a}\mathfrak{d}\mathfrak{p}$ lying between $\mathfrak{d}\mathfrak{p}$ and $\mathfrak{a}\mathfrak{p}$. They asked whether $\mathfrak{a}\mathfrak{d}\mathfrak{p}$ coincides with either of these two cardinals. We prove in ZFC that $\mathfrak{a}\mathfrak{d}\mathfrak{p}=\mathfrak{d}\mathfrak{p}$. We define a dual variant ${\mathfrak{a}\mathfrak{d}\mathfrak{p}}_2$ and show that ${\mathfrak{a}\mathfrak{d}\mathfrak{p}}_2=\mathfrak{a}\mathfrak{p}$. We further study the relation between $\mathfrak{a}\mathfrak{p}$ and the weakened $Q$-space cardinals. We introduce a tree analogue $\mathfrak{a}\mathfrak{t}$ of $\mathfrak{a}\mathfrak{p}$ and prove ${\mathfrak{q}}_1\le \mathfrak{a}\mathfrak{t}\le {\mathfrak{q}}_{2\frac{1}{2}}$, hence $\mathfrak{a}\mathfrak{p}\le {\mathfrak{q}}_{2\frac{1}{2}}$. Assuming the Generalized Continuum Hypothesis, we construct ccc forcing extensions with $\mathfrak{a}\mathfrak{p}={\omega }_1<\mathfrak{a}\mathfrak{t}={\mathfrak{q}}_{2\frac{1}{2}}=\mathfrak{c}$, so $\mathfrak{a}\mathfrak{p}<\mathfrak{a}\mathfrak{t}$ is consistent with ZFC.