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

There are several competing definitions of a "proper function between two topological spaces if the preimage of every compact if it is continuous and ; that is if it is a continuous closed map is compact is locally compact and Hausdorff. is compact.

a finite set.

which completes the proof.

is locally compact Hausdorff then proper is equivalent to

are metric spaces

Properties

• Every continuous map from a compact space to a Hausdorff space is both proper and closed.

Every surjective proper map is a compact covering map.

• is called a

• A topological space is compact if and only if the map from that space to a single point is proper.

• is a compactly generated Hausdorff space (this includes Hausdorff spaces that are either first-countable or locally compact is closed.

Generalization

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

See also

Citations

References

• , esp. section C3.2 "Proper maps"

• , esp. p.90 "Proper maps" and the Exercises to Section 3.6.