Some new results on $Δ$-spaces

TL;DR

This work studies $\Delta$-spaces, spaces where every decreasing sequence of subsets with empty intersection can be refined by a decreasing sequence of open sets with empty intersection. It proves in ZFC that every $T_3$ countably compact $\Delta$-space is compact, using stationarity arguments, $\sigma$-compactness, and the Alexandrov one-point compactification to derive a contradiction. It then shows a dichotomy for crowded Baire $T_1$ $\Delta$-spaces: either there is an isolated point, or a crowded Baire irresolvable subspace exists, which, by known set-theoretic results, yields an inner model with a measurable cardinal. Finally, it derives a general bound $|X|<o(X)$ when $X\in\Delta$ and $\operatorname{cf}(o(X))>\omega$, improving earlier results and connecting topological properties of $\Delta$-spaces with large-cardinal phenomena. Together, the results provide both pure topological insights and links to set-theoretic strength, with implications for $C_p(X)$ and the structure of open sets.

Abstract

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.

Theorems & Definitions (11)

• Definition 2.1
• Lemma 2.2
• proof
• Theorem 2.3
• proof
• Theorem 3.1
• proof
• Corollary 3.2
• Theorem 4.1
• proof