# Proper map

{{#invoke:Hatnote|hatnote}}

In mathematics, a continuous 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 pre-image of every point in *Y* is compact. The two definitions are equivalent if Y is compactly generated 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 continuous 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 Template:Harv.

## See also

## References

- {{#invoke:citation/CS1|citation

|CitationClass=citation }}

- {{#invoke:citation/CS1|citation

|CitationClass=citation }}, esp. section C3.2 "Proper maps"

- {{#invoke:citation/CS1|citation

|CitationClass=citation }}, 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.

- ↑ {{#invoke:Citation/CS1|citation |CitationClass=journal }}