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.

Definition

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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   is compact 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   is compact. The two definitions are equivalent if   is locally compact and Hausdorff.

Partial proof of equivalence

Let   be a closed map, such that   is compact (in  ) for all   Let   be a compact subset of   It remains to 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 every   there exists a finite subset   such that   The set   is closed in   and its image under   is closed in   because   is a closed map. Hence the set   is open in   It follows that   contains the point   Now   and because   is assumed to be compact, there are finitely many points   such that   Furthermore, the set   is a finite union of finite sets, which makes   a finite set.

Now it follows that   and we have found a finite subcover of   which completes the proof.

If   is Hausdorff and   is locally compact Hausdorff then proper is equivalent to universally closed. A map is universally closed if for any topological space   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   and   are metric spaces is as follows: we say an infinite sequence of points   in a topological space   escapes to infinity if, for every compact set   only finitely many points   are in   Then a continuous map   is proper if and only if for every sequence of points   that escapes to infinity in   the sequence   escapes to infinity in  

Properties

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  • 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.
    • A map   is called a compact covering if for every compact subset   there exists some compact subset   such that  
  • A topological space is compact if and only if the map from that space to a single point is proper.
  • If   is a proper continuous map and   is a compactly generated Hausdorff space (this includes Hausdorff spaces that are either first-countable or locally compact), then   is closed.[2]

Generalization

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

See also

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  • Almost open map – Map that satisfies a condition similar to that of being an open map.
  • Open and closed maps – A function that sends open (resp. closed) subsets to open (resp. closed) subsets
  • Perfect map – Continuous closed surjective map, each of whose fibers are also compact sets
  • Topology glossary – Mathematics glossary

Citations

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  1. ^ Lee 2012, p. 610, above Prop. A.53.
  2. ^ Palais, Richard S. (1970). "When proper maps are closed". Proceedings of the American Mathematical Society. 24 (4): 835–836. doi:10.1090/s0002-9939-1970-0254818-x. MR 0254818.

References

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