New results about Q and $\Delta$-spaces

Abstract

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.