New results about Q and $Δ$-spaces

A topological space $X$ is called a $Q$-space if every subset of $X$ is a $G_\delta$-set, and $X$ is a $\Delta$-space if for any decreasing sequence $\{D_n : n \in\omega\}$ of subsets of $X$ with empty intersection there is a decreasing sequence $\{U_n : n \in \omega\}$ of open sets with empty intersection such that $D_n \subseteq U_n$ for all $n \in\omega$. Our main result shows that the following statements are equiconsistent: (1) There exists a measurable cardinal; (2) There exists a crowded Baire $T_1$ $\Delta$-space; (3) There exists a crowded Baire $T_4$ $Q$-space; (4) There exists a $T_1$ $\Delta$-space admitting a strictly positive probability measure vanishing on points; (5) There exists a $T_3$ $Q$-space admitting a strictly positive probability measure vanishing on points. This provides complete answers to some problems and partial answers to other problems that have recently appeared in the literature. We also prove a new result concerning Lindel\"of $Q$-spaces: if $X$ is a $T_3$ Lindel\"of $Q$-space with $w(X)\leq \mathfrak c$, then $|X|<\operatorname{cf}(\mathfrak c)$. This yields a number of nonexistence results for large Lindel\"of, locally compact, compact, and countably compact $Q$-spaces.