Borel isomorphism between a non-separable metric and Polish?

Question

Could a non-separable metric space (uncountable) be Borel-isomorphic to a compact Polish space (uncountable too)?

Or, put differently, what would be sufficient conditions on this non-separable metric to be Borel isomorphic to a compact Polish space?

Answer 1

No, it cannot. Indeed, assume that $X$ is a non-separable metric space and $f:X\to K$ is a Borel isomorphism between $X$ and a Polish compact space $K$. Since the space $X$ is non-separable and metric, it contains an uncountable closed discrete (even $\varepsilon$-separated for some $\varepsilon>0$) subset $A$. Then $f(A)$ is an uncountable Borel subset of the space $K$. The Perfect Set Theorem for Borel Sets (Alexandrov, Hausdorff [13.6, Kech]) claims each Borel subset of a Polish space either is countable or else it contains a Cantor set. Thus $f(A)$ contains a Cantor set $C$. Let $B=f^{-1}(C)$. Since each subset of $B$ is Borel (in $X$) then each subset of $C$ is Borel (in $C$), which is a contradiction.

References

[Kech] Alexander S. Kechris. Classical Descriptive Set Theory , Springer, 1995.