A MAD Q-set
Fundamenta Mathematicae, Tome 178 (2003) no. 3, p. 271.

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A MAD (maximal almost disjoint) family is an infinite subset of the infinite subsets of ω = 0,1,2,... such that any two elements of intersect in a finite set and every infinite subset of ω meets some element of in an infinite set. A Q-set is an uncountable set of reals such that every subset is a relative G δ -set. It is shown that it is relatively consistent with ZFC that there exists a MAD family which is also a Q-set in the topology it inherits as a subset of P ( ω ) = 2 ω . 
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Arnold W. Miller. A MAD Q-set. Fundamenta Mathematicae, Tome 178 (2003) no. 3, p. 271. https://geodesic-test.mathdoc.fr/item/FUNDAM_2003__178_3_282884/