On the number of $Q$-points

Abstract

We show that, up to isomorphism, the number of $Q$-points is either finite, $2^{\mathfrak{d}}$ or $2^{\mathfrak{c}}$. This answers a question asked by Borodulin-Nadzieja, Martínez-Celis, Morawski and Świerczyńska, and by Halbeisen and the authors. We also show that under mild hypotheses, the existence of infinitely many $Q$-points implies the existence of non-atomic $Q$-measures, and of $2^{\mathfrak{c}}$-many Tukey-top $Q$-points, strengthening results of Raghavan and of Borodulin-Nadzieja et al..