Saved in:
| Main Authors: | , , , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2510.04242 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866916990861443072 |
|---|---|
| author | Juhász, I. van Mill, J. Soukup, L. Szentmiklóssy, Z. |
| author_facet | Juhász, I. van Mill, J. Soukup, L. Szentmiklóssy, Z. |
| contents | A topological space $X$ is a $Δ$-space (or $X \in Δ$) if for any decreasing sequence $\{A_n : n < ω\}$ of subsets of $X$ with empty intersection there is a (decreasing) sequence $\{U_n : n < ω\}$ of open sets with empty intersection such that $A_n \subset U_n$ for all $n < ω$. In this note we prove the following results concerning $Δ$-spaces.
1) Every $T_3$ countably compact $Δ$-space is compact.
2) If there is a $T_1$ crowded Baire $Δ$-space then there is an inner model with a measurable cardinal.
3) If $X \in Δ$ and $cf \big(o(X) \big) > ω$ then $|X| < o(X)$. (Here $o(X)$ is the number of open subsets of $X$.)
The first two of these provide full and/or partial solutions to problems raised in the literature, while the third improves a known result. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_04242 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Some new results on $Δ$-spaces Juhász, I. van Mill, J. Soukup, L. Szentmiklóssy, Z. General Topology 54A25 A topological space $X$ is a $Δ$-space (or $X \in Δ$) if for any decreasing sequence $\{A_n : n < ω\}$ of subsets of $X$ with empty intersection there is a (decreasing) sequence $\{U_n : n < ω\}$ of open sets with empty intersection such that $A_n \subset U_n$ for all $n < ω$. In this note we prove the following results concerning $Δ$-spaces. 1) Every $T_3$ countably compact $Δ$-space is compact. 2) If there is a $T_1$ crowded Baire $Δ$-space then there is an inner model with a measurable cardinal. 3) If $X \in Δ$ and $cf \big(o(X) \big) > ω$ then $|X| < o(X)$. (Here $o(X)$ is the number of open subsets of $X$.) The first two of these provide full and/or partial solutions to problems raised in the literature, while the third improves a known result. |
| title | Some new results on $Δ$-spaces |
| topic | General Topology 54A25 |
| url | https://arxiv.org/abs/2510.04242 |