# Continuous, proper, injective map into first-countable space is homeomorphism onto image

Theorem: Let $X, Y$ be topological spaces where $Y$ is first-countable and Hausdorff and let $f: X\rightarrow Y$ be an injective, continuous map which is proper. Then $f$ is a homeomorphism into it's image.

Proof: The only thing left to show is that $f^{-1}: f(X) \rightarrow X$ is continuous. Since $Y$ is first-countable the subspace $f(X) \subseteq Y$ is first-countable too. In particular $f(X)$ is a compactly generated space. Therefore it is enough to show that for each compact subset $K\subseteq f(X)$ the restriction $f^{-1}|_K: K\rightarrow f^{-1}(K)$ is continuous.

As a compact subset of $f(X)$ the set $K$ is also compact in $Y$. Since $f$ is proper $f^{-1}(K)\subseteq X$ is compact. So $f|_{f^{-1}(K)}: f^{-1}(K) \rightarrow K$ is a continuous and bijective map on a compact space into a Hausdorff space. It is well-known that this already implies that $f|_{f^{-1}(K)}$ is a homeomorphism. In particular the inverse function $f^{-1}|_K: K\rightarrow f^{-1}(K)$ is continuous. This proves the theorem.

Is this proof correct?