I am reading T. Szamuely's book on Galois groups and fundamental groups. As preparation to the algebraic case, he recalls the topological case. So I am wondering if a surjective local homeomorphism $f$ from some connected space $X$ to $\mathbb{R}^n$ is necessarily a covering, in which case it would be bijective since $\mathbb{R}^n$ is simply connected.

What about the differentiable case?