Co-analytic $Q$-sets A subset $A\subseteq \mathbb{R}$ is said to be a $Q$-set if every subset $B\subseteq A$ is $F_\sigma$ wrt the subspace topology on $A$. For example $\mathbb{Q}$ is a $Q$-set. The first time I have seen this definition is in:
Balogh, Zoltán, There is a $Q$-set space in ZFC, Proc. Am. Math. Soc. 113, No. 2, 557-561 (1991).
Now my questions are:

*

*Can we prove in $\mathtt{ZF}+\mathtt{DC}$, where $\mathtt{DC}$ stands for dependent choice, that every co-analytic $Q$-set is countable?

*Is the descriptive complexity (with connections with set-theoretic hypothesis) of these sets been studied in any paper/book/thesis?

Thanks!
 A: I do not know about full references but if you haven't seen it yet there is a nice discussion of this topic in Miller's https://people.math.wisc.edu/~miller/res/dstfor.pdf. (Incidentally this may be the most fun math book to read). See in particular sections 2-5.
In particular, Theorem 5.1 states that Martin's Axiom implies every second countable Hausdorff space of cardinality less than the continuum is a Q-set. If $V=L$ there are $\Pi^1_1$ sets $A \subseteq \mathbb R$ of size $\aleph_1$ so that in any $\aleph_1$-preserving forcing remain uncountable and no new elements are added. Forcing martin's Axiom over $L$ with ccc forcing (so the standard way) will therefore give a consistent example of an $\aleph_1$-sized co-analytic Q-set.
Also Corollary 3.2 can be used to answer your first question. This corollary states that any second countable space which contains a perfect set is not Q (actually it uses a weaker hypothesis to make a stronger conclusion). Staring at the proof of the relevant results towards proving this corollary you need to convince yourself that DC is enough. Given this, if every set $A \subseteq \mathbb R$ contains a perfect set (for instance in the Solovay model) then no uncountable set of reals can be a Q-set.
Putting these two results together also we get (modulo an inaccessible) that the existence of a projective uncountable Q-set is independent of ZFC.
I do not know if anyone has thought about these definability issues further, though I would check the references of the Miller book to start.
