View source for Proper map

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{{MSC|54C05}}

Let $X$ and $Y$ be two [[Topological space|topological spaces]]. A [[Continuous mapping|continuous map]] $f:X\to Y$ is called ''proper'' if $f^{-1} (K)$ is [[Compact space|compact]] for every $K\subset Y$ compact.

When $X$ is [[Hausdorff space|Hausdorff]] and $Y$ [[Locally compact space|locally compact]] the properness of $f$ is equivalent to the requirement that $f^{-1} (\{y\})$ is compact for every $y\in Y$.

If $X$ is a compact space and $Y$ is Hausdorff, then any continuous map $f:X\to Y$ is proper and [[Closed mapping|closed]].