Let $X,Y$ be complete separable metric spaces, with $X$ locally compact, and $C(X,Y)$ the space of continuous functions from $X$ to $Y$, equipped with the topology of uniform convergence on compact sets. If I am not mistaken, $C(X,Y)$ is Polish.
Let $S \subset C(X,Y)$ be the set of functions in $C(X,Y)$ which are surjective.
Is $S$ Borel? If not, what can we say about its complexity?
If $X$ is compact, it is easy to show that $S$ is closed. But the locally compact case seems harder.
In particular, I have in mind something like $X = \mathbb{R} \times [0,1]^2$ and $Y = \mathbb{R}^d$.
Thanks!