A characterization of proper maps

Question

Let $M,N$ topological spaces, $F:M\to N$ a continuos map is proper if (by definitios) $F^{-1}(K)$ is compact for any $K\subseteq N$ compact. Suppose now that $M,N$ are locally compact and Hausdorff. Then the following are equivalent:

1)$F$ is proper

2)$F$ is closed and $F^{-1}(q)$ is compact for any $q\in N$

I'm able to prove the implication 1)$\Rightarrow$2), but not the converse.

Let $K$ a compact subset of $N$, then it's closed (beacuse N is Hausdorff), so an idea could be to prove that $F^{-1}(K)$ is contained in some compact subset. A way can be these (it's not suggested by the exercise, it's only a my perhaps wrong idea): Prove that for each $P\in N$ there is some $V_P$ compact neighborhood s.t its preimage is compact (one i think should use here the local-compactness property and the fact that F^{-1}(P) is compact). $K$ is contained a finite union of neighborhoods of such kind (by compactness of K), then $F^{-1}(K)$ is contained in a closed union of compacts, which are the preimage of the neighbouroods

Answer 1

I’ll use theorems from “General topology” by Ryszard Engelking (Heldermann Verlag, Berlin, 1989). In fact, you are dealing with basic properties of perfect maps, which are exactly continuous maps $f$ from a Hausdorff space $M$ to a space $N$, satisfying 2). That is you have proved that each proper map between given $M$ and $N$ is perfect. The converse implication (without any restrictions imposed on spaces $M$ and $N$) is the subject of

In our case the equivalence between proper and perfect maps also follows from

Also if we already have a perfect map then it suffices to require that only one of the spaces $M$ and $N$ is locally compact due to preservation of (local) compactness and Hausdorffness by perfect maps.