Decomposing $\mathbf{\Pi}^1_1$ sets into closed sets It is well known that every $\mathbf{\Pi}^1_1$-set is a union of $\aleph_1$-many Borel sets. I wonder whether it can be improved under certain reasonable set theory axioms assumption.
For example, assuming  $ZFC+CH$, then it is trivially true that every set is a union of $\aleph_1$-many closed sets. But this seems heavily depends on $CH$ since  if $ZFC+\neg CH+MA$, then there is a lightface $\Pi^0_2$-set which cannot be a union of $\aleph_1$-many closed sets.
So my question is: is it consistent with $ZFC+\neg CH$ that every  $\mathbf{\Pi}^1_1$-set is a union of $\aleph_1$-many closed sets?
 A: Every ${\bf\Pi}_1^1$ set is the union of $\aleph_1$ Borel sets, so we only have to make sure that every Borel set is the union of $\aleph_1$ closed sets.
But every Borel set is analytic and thus a continuous image of the Baire space $\omega^\omega$.
Now the Baire space is the union of $\mathfrak d$ (the dominating number) compact sets.  Continuous images of compact sets are again compact and hence every analytic set is the union of $\mathfrak d$ compact (and hence closed) sets.
It is well known that $\mathfrak d=\aleph_1$ is consistent with the failure of CH.
This happens for instance in the so called Sacks model.

Edit:  I forgot to mention that it was Paul Larson who turned my attention to this question.  Paul conjectured (or should I say, was convinced) that in the model obtained by forcing with a large countable support product of Sacks forcing over a model of CH every ${\bf\Pi}_1^1$ set is the union of $\aleph_1$ closed sets, 
which is true, since $\mathfrak d$ is $\aleph_1$ in that model.
A: There is a theorem of my teacher Steve Jackson which says that assuming $ZFC + AD^{L(\mathbb{R})}$ every projective set is $\aleph_{\omega}$-Borel. So in particular this holds for $\Pi^1_1$ sets. The proof uses the theory of descriptions and every other technical tool from descriptive set theory (homogeneous trees, scales,...). Also, with respect to $MA$ and $CH$, $AD$ can't decide them, so maybe that result might be what you're looking for, I'm not sure. You can find the result in this survey of Jackson "A survey of Determinacy" somewhere in the end of the paper. 
