# Proper map

In mathematics, a function between topological spaces is called **proper** if inverse images of compact subsets are compact. In algebraic geometry, the analogous concept is called a proper morphism.

## Definition

A function *f* : *X* → *Y* between two topological spaces is **proper** if the preimage of every compact set in *Y* is compact in *X*.

There are several competing descriptions. For instance, a continuous map *f* is proper 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. For a proof of this fact see the end of this section. More abstractly, *f* is proper if *f* is universally closed, i.e. if for any topological space *Z* the map

*f*× id_{Z}:*X*×*Z*→*Y*×*Z*

is closed. These definitions are equivalent to the previous one if *X* is Hausdorff and *Y* is locally compact Hausdorff.

An equivalent, possibly more intuitive definition when *X* and *Y* are metric spaces is as follows: we say an infinite sequence of points {*p*_{i}} in a topological space *X* **escapes to infinity** if, for every compact set *S* ⊂ *X* only finitely many points *p*_{i} are in *S*. Then a continuous map *f* : *X* → *Y* is proper if and only if for every sequence of points {*p*_{i}} that escapes to infinity in *X*, {*f*(*p*_{i})} escapes to infinity in *Y*.

This last sequential idea looks like being related to the notion of sequentially proper, see a reference below.

### Proof of fact

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.

## Properties

- 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
*f*:*X*→*Y*is a proper continuous map and*Y*is a compactly generated Hausdorff space (this includes Hausdorff spaces which are either first-countable or locally compact), then*f*is closed.^{[1]}

## Generalization

It is possible to generalize the notion of proper maps of topological spaces to locales and topoi, see (Johnstone 2002).

## See also

## References

- 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*, N. Carolina: Booksurge, ISBN 1-4196-2722-8, esp. p. 90 "Proper maps" and the Exercises to Section 3.6. - Brown, R. "Sequentially proper maps and a sequential compactification", J. London Math Soc. (2) 7 (1973) 515-522.

- ↑ Palais, Richard S. (1970). "When proper maps are closed" (PDF).
*Proc. Amer. Math. Soc*.**24**: 835–836. doi:10.1090/s0002-9939-1970-0254818-x.