= Proper maps extend to one-point compactifications
{title2=$\alpha f(\infty_X)=\infty_Y$}
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>.
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