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.

DefinitionEdit

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.

PropertiesEdit

GeneralizationEdit

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

See alsoEdit

ReferencesEdit

  • Bourbaki, Nicolas (1998). General topology. Chapters 5–10. Elements of Mathematics. Berlin, New York: Springer-Verlag. ISBN 978-3-540-64563-4. MR 1726872.
  • Johnstone, Peter (2002). Sketches of an elephant: a topos theory compendium. Oxford: Oxford University Press. ISBN 0-19-851598-7., esp. section C3.2 "Proper maps"
  • Brown, Ronald (2006). Topology and groupoids. North Carolina: Booksurge. ISBN 1-4196-2722-8., esp. p. 90 "Proper maps" and the Exercises to Section 3.6.
  • Brown, Ronald (1973). "Sequentially proper maps and a sequential compactification". Journal of the London Mathematical Society. 2. 7: 515–522.