Corollary

Let $X$ be a Hausdorff space, $Y$ a topological space, $f : X \to Y$ a proper mapping. Then for $X$ to be compact (resp. locally compact) it is necessary and sufficient that $f(X)$ is compact (resp. locally compact), and it is sufficient that $Y$ is compact (resp. locally compact).

f_z : X \to f(X) is proper [no. 1, Proposition 3 a)] it follows that X is compact (resp. locally compact) by reason of Proposition 6 of no. 2 and Proposition 7 of no. 3. Finally, if Y is compact (resp. locally compact) then so is f(X) which is closed in Y (no. 1, Proposition 1 and , no. 7, Proposition 13)