Almost Disjointness Principles and $Q$-Space Cardinals

TL;DR

This paper investigates relationships between various set-theoretic cardinals related to disjointness and $Q$-spaces, proving equalities and inequalities, and constructing models with specific cardinal configurations.

Contribution

It establishes that $rak{adp}=rak{dp}$ and $rak{adp}_2=rak{ap}$ within ZFC, introduces a tree analogue $rak{at}$, and shows the consistency of $rak{ap}<rak{at}$ under GCH.

Findings

$rak{adp}=rak{dp}$ in ZFC

$rak{adp}_2=rak{ap}$ in ZFC

Consistent separation $rak{ap}<rak{at}$ with GCH

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 $\mathfrak q_1\leq\mathfrak{at}\leq\mathfrak q_{2\frac{1}{2}}$, 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.