Definition:Proper Mapping

Definition
Let $X$ and $Y$ be topological spaces.

A mapping $f: X \to Y$ is proper for every compact subset $K \subset Y$, its preimage $f^{-1} \left({K}\right)$ is also compact.

Also defined as
A proper mapping is sometimes defined as a closed mapping such that the preimage of every point is compact.

Also known as
A proper mapping can also be referred to as a proper map or a proper function.

standardises on mapping rather than map, and reserves the term function for numbers.

Also see

 * Proper Map from Hausdorff to Compact Hausdorff Space is Continuous