Some new results on $\Delta$-spaces

A topological space $X$ is a $\Delta$-space (or $X \in \Delta$) if for any decreasing sequence $\{A_n : n < \omega\}$ of subsets of $X$ with empty intersection there is a (decreasing) sequence $\{U_n : n < \omega\}$ of open sets with empty intersection such that $A_n \subset U_n$ for all $n < \omega$. In this note we prove the following results concerning $\Delta$-spaces. 1) Every $T_3$ countably compact $\Delta$-space is compact. 2) If there is a $T_1$ crowded Baire $\Delta$-space then there is an inner model with a measurable cardinal. 3) If $X \in \Delta$ and $cf \big(o(X) \big) > \omega$ 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.