Corollary

Let X, X' be two locally compact spaces, and let Y (resp. Y') be the compact space obtained by adjoining a point at infinity $\omega$ (resp. $\omega'$) to X (resp. X') (§ 9, no. 8). Then a continuous mapping $f : X \to X'$ is proper if and only if its extension $\overline{f} : Y \to Y'$, such that $\overline{f}(\omega) = \omega'$, is continuous.

By Proposition 7, f is proper if and only if, for each compact subset K' of X', $\overline{f}^{-1}(X' - K') = X - \overline{f}^{-1}(K')$ is the complement of a compact subset of X; by the definition of the neighbourhoods of $\omega$ (resp. $\omega'$) in Y (resp. Y') this is so if and only if $\overline{f}$ is continuous at $\omega$.