Suppose that $f:X\to Y$ is a Borel map from separable metric space $X$ to a $T_3$ space $Y$.
Does there always exist a Polish space $\tilde X \supseteq X$ and $T_3$ space $\tilde Y\supseteq Y$ and an extension $\tilde f:\tilde X\to \tilde Y$ of $f$ in the sense that $\tilde f$ is Borel and $\tilde f(x) = f(x)$ for $x\in X$?
Context: I'm trying to consider spaces which are Borel images of separable metric spaces. However, the theory seems to be available for Polish spaces and not separable metric spaces. That's why I'm trying to show that a Borel image of separable metric space is a subspace of a Borel image of a Polish space, for which I know what to do. I've tried to do this by exhibiting a Borel map $g:\tilde X\to X$, however, thats only really available if $X$ is an analytic set. Thus I think this might be approachable by extending both the domain and codomain, hence the question.