# Proper map on from compact manifolds

Let $X$ be a compact manifold, every map $f: X \longrightarrow Y$ is proper.
In $Y$, compact sets are closed (assuming $Y$ is Hausdorff). $f$ is continuous, so the inverse image of a closed set is closed. But a closed subset of a compact (Hausdorff) space is compact. So the inverse image of a compact set is compact.