A Mad Q-set
  A MAD (maximal almost disjoint) family is an infinite subset A of the infinite subsets of {0,1,2,..} such that any two elements of A intersect in a finite set and every infinite subset of {0.1.2...} meets some element of $\aa$ in an infinite set. A Q-set is an uncountable set of reals such that every subset is a relative G-delta 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 in inherits a subset of the Power set of {0,1,2,..}, ie the Cantor set. 
Arnold W. Miller