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).