5
$\begingroup$

Let $X$ and $Y$ be two locally compact Hausdorff spaces, and let $X^+$ and $Y^+$ denote the one point compactifications of $X$ and $Y$, respectively. Let $f: X\rightarrow Y$ be a continuous function and let $f^+: X^+ \rightarrow Y^+$ be the obvious extension of $f$. Show that $f^+$ is continuous if and only if $f$ is proper.

By proper, I mean that for every compact subset $U$ of $Y$, the preimage of $U$ is compact.

Any help would be appreciated.

1 Answers 1