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 $(1)\Rightarrow(2)$.

vaguelyrelated. I'll think about your questions (I am rusty though). To me, the general topic started with the eternal claim of many of my students:if a set is not open then it's closed (and vice versa)). Thus at first, I introducedstudent spaces. Later I saw an exercise for Hausdorff spaces in Kelly's classic, these spaces were called thedoor spaces(I guess because a door is either open or closed). $\endgroup$ – Wlod AA Aug 13 '18 at 19:11