Proper map

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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[edit]

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[edit]

  • 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[edit]

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

See also[edit]

Citations[edit]

  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[edit]