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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 $(1)\Rightarrow(2)$.

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  • $\begingroup$ Taras, there is a related paper by wh & A.Mishchenko, published in Bull.Acad.Polon.Sci. in 1966 (or 1967?). It's in Russian (but I don't think you'd have any problem with this). $\endgroup$
    – Wlod AA
    Commented Aug 13, 2018 at 10:45
  • $\begingroup$ @WlodAA Thank you for the comment. I looked at the review of this paper in Zbl (zbmath.org/scans/241/357.gif) and found no confiramtions of this conjecture. There is however a theorem that a compact Hausdorff space is perfectly normal if all its open sets are Souslin. What is the relation of Souslinness of open set to Borelness of all subsets? $\endgroup$
    – Taras Banakh
    Commented Aug 13, 2018 at 12:46
  • $\begingroup$ Taras, I should have said vaguely related. 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 introduced student spaces. Later I saw an exercise for Hausdorff spaces in Kelly's classic, these spaces were called the door spaces (I guess because a door is either open or closed). $\endgroup$
    – Wlod AA
    Commented Aug 13, 2018 at 19:11

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