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 $\mathfrak{adp}$ 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 $\mathfrak q_1\leq\mathfrak{at}\leq\mathfrak q_{2\frac{1}{2}}$, hence $\mathfrak{ap}\leq\mathfrak q_{2\frac{1}{2}}$. Assuming the Generalized Continuum Hypothesis, we construct ccc forcing extensions with $\mathfrak{ap}=\omega_1<\mathfrak{at}=\mathfrak q_{2\frac{1}{2}}=\mathfrak c$, so $\mathfrak{ap}<\mathfrak{at}$ is consistent with ZFC.