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