Almost Disjointness Principles and $Q$-Space Cardinals
Fuente:
arXiv
Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2026
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866913115413676032 |
|---|---|
| author | Rodrigues, Vinicius de Oliveira |
| author_facet | Rodrigues, Vinicius de Oliveira |
| contents | 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 $\mathfrak{ap}=ω_1<\mathfrak{at}=\mathfrak q_{2\frac{1}{2}}=\mathfrak c$, so $\mathfrak{ap}<\mathfrak{at}$ is consistent with ZFC. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_11326 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Almost Disjointness Principles and $Q$-Space Cardinals Rodrigues, Vinicius de Oliveira Logic General Topology Primary 03E17, 03E05. Secondary 03E35, 54A35, 54D10 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 $\mathfrak{ap}=ω_1<\mathfrak{at}=\mathfrak q_{2\frac{1}{2}}=\mathfrak c$, so $\mathfrak{ap}<\mathfrak{at}$ is consistent with ZFC. |
| title | Almost Disjointness Principles and $Q$-Space Cardinals |
| topic | Logic General Topology Primary 03E17, 03E05. Secondary 03E35, 54A35, 54D10 |
| url | https://arxiv.org/abs/2605.11326 |