# Definition:Proper Mapping

## Definition

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

A mapping $f: X \to Y$ is proper if and only if 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.

$\mathsf{Pr} \infty \mathsf{fWiki}$ standardises on mapping rather than map, and reserves the term function for numbers.