Proper maps extend to one-point compactifications ($\alpha f({\infty }_X)={\infty }_Y$)

For locally compact noncompact Hausdorff spaces, a continuous proper map $f:X\to Y$ extends continuously to their one-point compactifications by the displayed rule. The inverse image of $\alpha Y\setminus K$ is $\alpha X\setminus f^{-1}(K)$, a neighborhood of infinity because the inverse image of a compact set is compact. The extension itself is proper, since it is continuous from a compact space to a Hausdorff space. A nonproper map can fail to admit any continuous extension: projection ${\mathbb{R}}^2\to \mathbb{R}$ sends two vertical sequences escaping to the same point at infinity to two distinct constant sequences, incompatible with continuity at that point.