Borel measurability of "closest point selection" Let $(Y,d)$ be a metric space, and let $X \subset Y$. Does there exist a Borel measurable function $\gamma: Y \to X$ such that, for all $y \in Y$, $d(y,\gamma(y)) \leq 2 \inf\{d(y,x) : x \in X\}$? I would be interested in such a selection function with the assumption that $X$ is Borel, or even closed, but I would prefer an answer that works for arbitrary $X$.

If $X$ is separable, then yes. Take a countable dense subset $\left\{x_n\right\}_n$.
Define $f:Y\to\mathbb{N}$ by $f(x)=\operatorname{min}\left\{n:d(x,x_n)<2d(x,X)\right\}$. Then $f$ is measurable (exercise), and the map $x:\mathbb{N}\to X$, $x(n)=x_n$ is trivially measurable as well, so $\gamma=xf$ works.
My guess is that this could be false for non-separable $X$, but I don't have any counter-example at hand.
