A $Q$-set is a set of reals every subset of which is a relative $G_\delta$. We investigate the combinatorics of $Q$-sets and discuss a question of Miller and Zhou on the size $\qq$ of the smallest set of reals which is not a $Q$-set. We show in particular that various natural lower bounds for $\qq$ are consistently strictly smaller than $\qq$.

Keywords:

$Q$-set, cardinal invariants of the continuum, pseudointersection number, $\MA$($\sigma$-centered), Dow's principle, almost disjoint family, almost disjointness principle, iterated forcing $Q$-set, cardinal invariants of the continuum, pseudointersection number, $\MA$($\sigma$-centered), Dow's principle, almost disjoint family, almost disjointness principle, iterated forcing

MSC Classifications:

03E05, 03E35, 54A35 show english descriptions Other combinatorial set theory
Consistency and independence results
Consistency and independence results [See also 03E35] 03E05 - Other combinatorial set theory
03E35 - Consistency and independence results
54A35 - Consistency and independence results [See also 03E35]

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