# Definition:Proper Mapping

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## 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.