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 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
- 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