Proper map

In mathematics, a function between topological spaces is called proper if inverse images of compact subsets are compact.

[1] In algebraic geometry, the analogous concept is called a proper morphism.

There are several competing definitions of a "proper function".

Some authors call a function

between two topological spaces proper if the preimage of every compact set in

Other authors call a map

proper if it is continuous and closed with compact fibers; that is if it is a continuous closed map and the preimage of every point in

The two definitions are equivalent if

is locally compact and Hausdorff.

be a closed map, such that

It remains to show that

be an open cover of

this is also an open cover of

Since the latter is assumed to be compact, it has a finite subcover.

there exists a finite subset

is a closed map.

is assumed to be compact, there are finitely many points

is a finite union of finite sets, which makes

and we have found a finite subcover of

is locally compact Hausdorff then proper is equivalent to universally closed.

A map is universally closed if for any topological space

is Hausdorff, this is equivalent to requiring that for any map

be closed, as follows from the fact that

is a closed subspace of

An equivalent, possibly more intuitive definition when

are metric spaces is as follows: we say an infinite sequence of points

escapes to infinity if, for every compact set

Then a continuous map

is proper if and only if for every sequence of points

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