There are several competing descriptions. For instance, a continuous map f is proper if it is closed with compact fibers, i.e. if it is a closed map and the preimage of every point in Y is compact. The two definitions are equivalent if Y is locally compact and Hausdorff.
Partial proof of equivalence
Let be a closed map, such that is compact (in X) for all . Let be a compact subset of . We will show that is compact.
Let be an open cover of . Then for all this is also an open cover of . Since the latter is assumed to be compact, it has a finite subcover. In other words, for all there is a finite set such that . The set is closed. Its image is closed in Y, because f is a closed map. Hence the set
is open in Y. It is easy to check that contains the point . Now and because K is assumed to be compact, there are finitely many points such that . Furthermore the set is a finite union of finite sets, thus is finite.
Now it follows that and we have found a finite subcover of , which completes the proof.
If X is Hausdorff and Y is locally compact Hausdorff then proper is equivalent to universally closed. A map is universally closed if for any topological space Z the map is closed. In the case that is Hausdorff, this is equivalent to requiring that for any map the pullback be closed, as follows from the fact that is a closed subspace of .
An equivalent, possibly more intuitive definition when X and Y are metric spaces is as follows: we say an infinite sequence of points in a topological space X escapes to infinity if, for every compact set only finitely many points are in S. Then a continuous map is proper if and only if for every sequence of points that escapes to infinity in X, the sequence escapes to infinity in Y.
- A topological space is compact if and only if the map from that space to a single point is proper.
- Every continuous map from a compact space to a Hausdorff space is both proper and closed.
- If is a proper continuous map and Y is a compactly generated Hausdorff space (this includes Hausdorff spaces that are either first-countable or locally compact), then f is closed.
- Bourbaki, Nicolas (1998). General topology. Chapters 5–10. Elements of Mathematics. Berlin, New York: Springer-Verlag. ISBN 978-3-540-64563-4. MR 1726872.
- Johnstone, Peter (2002). Sketches of an elephant: a topos theory compendium. Oxford: Oxford University Press. ISBN 0-19-851598-7., esp. section C3.2 "Proper maps"
- Brown, Ronald (2006). Topology and groupoids. North Carolina: Booksurge. ISBN 1-4196-2722-8., esp. p. 90 "Proper maps" and the Exercises to Section 3.6.
- Brown, Ronald (1973). "Sequentially proper maps and a sequential compactification". Journal of the London Mathematical Society. 2. 7: 515–522.