I am interested in characterizing compact topological spaces all of whose subsets are Borel. In this respect I have the following
Conjecture. For a compact Hausdorff space $X$ the following conditions are equivalent:
1) Each subsets of $X$ is Borel.
2) $X$ is scattered of countable scattered height.
Remark. The implication $(2)\Rightarrow(1)$ is true and can be proved by induction on the scattered height of $X$. So, it remains to prove $(1)\Rightarrow(2)$.