Does there exist an uncountable separable metric space $X$ such that every subset of $X$ is a Borel set?

Question

Is it consistent with ZFC that there exists an uncountable separable metric space $X$ such that every subset of $X$ is a Borel set?

If the continuum hypothesis holds, or more generally $2^{\aleph_{0}}<2^{\aleph_{1}}$ , then each uncountable separable metric space contains non-Borel sets since there are only $2^{\aleph_{0}}$ Borel sets. Therefore, in order for there to be an uncountable separable metric space $X$ where every subset of $X$ is Borel, we would need to at least have $2^{\aleph_{0}}=2^{\aleph_{1}}$. What other conditions are necessary in order for there to exist an uncountable separable metric space $X$ where every subset of $X$ is Borel?

Answer 1

Under Martin's Axiom plus the negation of CH, every set $X$ of reals of size $<\mathfrak c$ is a Q-set, which means that every subset of $X$ is an $F_\sigma$-set with respect to the subspace topology that $X$ inherits from the real line. For more about such sets, see Arnie Miller's chapter, "Special subsets of the real line" in the Handbook of Set-Theoretic Topology.