\begin{definition}[Definition:Proper Mapping]
Let $X$ and $Y$ be topological spaces.
A mapping $f: X \to Y$ is '''proper''' {{iff}} for every compact subset $K \subset Y$, its preimage $f^{-1} \left({K}\right)$ is also compact.
{{explain|Here and elsewhere: is it worth replacing "compact subset" with "compact subspace" wherever it appears, as "subset" implies just the set of elements -- "subspace" makes it clear that the topology imposed on that subset is important. I don't know whether this flies against the prevailing convention, but it would help to make the language of this complicated area, where extreme clarity is required, more easily understood.}}
\end{definition}