New results about Q and $\mathrm{\Delta }$-spaces

A topological space $X$ is called a $Q$-space if every subset of $X$ is a $G_{\delta }$-set, and $X$ is a $\mathrm{\Delta }$-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 $T_1$ $\mathrm{\Delta }$-space; (3) There exists a crowded Baire $T_4$ $Q$-space; (4) There exists a $T_1$ $\mathrm{\Delta }$-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 Lindelöf $Q$-spaces: if $X$ is a $T_3$ Lindelöf $Q$-space with $w(X)\le \mathfrak{c}$, then $\mathrm{\mid }X\mathrm{\mid }<\mathrm{cf}(\mathfrak{c})$. This yields a number of nonexistence results for large Lindelöf, locally compact, compact, and countably compact $Q$-spaces.