Theorem 2.6:

Suppose $X$ and $Y$ are locally compact Hausdorff spaces and $f: \widehat{X} \longrightarrow \widehat{Y}$ is a continuous map such that $f^{-1}(q) = \{p\}$, where $p$ and $q$ are as in the previous theorem, then the restriction of $f$ to $X$ is a proper map.