2026/07/01 by János Balázs Ivanyos, Ákos Székely · 1 voice
Mathematics · #math.GN #math.LO
A topological space \(X\) is called a \(Q\)-space if every subset of \(X\) is a \(Gδ\)-set, and \(X\) is a \(Δ\)-space if for any decreasing sequence \(\Dn : n ∈ω\\) of subsets of \(X\) with empty intersection there is a decreasing sequence \(\Un : n ∈ ω\\) of open sets with empty intersection such that \(Dn ⊆ Un\) for all \(n ∈ω\). Our main result shows that the following statements are equiconsistent: (1) There exists a measurable cardinal; (2) There exists a crowded Baire \(T1\) \(Δ\)-space; (3) There exists a crowded Baire \(T4\) \(Q\)-space; (4) There exists a \(T1\) \(Δ\)-space admitting a strictly positive probability measure vanishing on points; (5) There exists a \(T3\) \(Q\)-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 \(T3\) Lindelöf \(Q\)-space with \(w(X)≤ \mathfrak c\), then \(|X|<cf(\mathfrak c)\). This yields a number of nonexistence results for large Lindelöf, locally compact, compact, and countably compact \(Q\)-spaces.