Gespeichert in:
| Hauptverfasser: | , , , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2025
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| Schlagworte: | |
| Online-Zugang: | https://arxiv.org/abs/2510.04242 |
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Inhaltsangabe:
- 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.