On $Q$ Sets

William G. Fleissner, Arnold W. Miller
Proceedings of the American Mathematical Society, Vol. 78, No. 2. (Feb., 1980), pp. 280-284.


A $Q$ set is an uncountable set $X$ of the real line such that every subset of $X$ is an $F_\sigma$ relative to $X$. It is known that the existence of a $Q$ set is independent of and consistent with the usual axioms of set theory. We show that one cannot prove, using the usual axioms of set theory: 1. If $X$ is a $Q$ set then any set of reals of cardinality less than the cardinality of $X$ is a $Q$ set. 2. The union of a $Q$ set and a countable set is a $Q$ set.

Keywords: 03E25, 54A25, 54E30, $Q$ set, iterated forcing, pathological sets of reals, normal Moore space conjecture