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 subspace $K \subset Y$, its preimage $f^{-1} \sqbrk K$ 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.

Also see

• Proper Mapping from Hausdorff to Compact Hausdorff Space is Continuous

• Results about proper mappings can be found here.

Sources

• 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): proper map