Assume Martin's axiom plus $\neg CH$. It is well known, via almost disjoint forcing, that every set of reals of size less than continuum is an example of a metric space whose subsets are all relatively $F_{\sigma}$ and hence Borel. Now suppose (along with $MA + \neg CH$) that every subset of a metric space $X$ is Borel. Must every subset of $X$ be $F_{\sigma}$?
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2$\begingroup$ Without MA + not CH, it is consistent that there is a set of reals $X$ such that every subset of $X$ is relatively Borel, but not every subset is relatively $F_\sigma$. This is due to Arnie Miller. But I guess this is not what you were looking for? $\endgroup$– Paul McKenneyCommented Nov 26, 2015 at 13:14
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Answer copied from the comments:
Without MA + not CH, it is consistent that there is a set of reals $X$ such that every subset of $X$ is relatively Borel, but not every subset is relatively $F_\sigma$. This is due to Arnie Miller. – Paul McKenney