Proper map

Short description: Map between topological spaces with the property that the preimage of every compact is compact

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.

Definition

There are several competing definitions of a "proper function". Some authors call a function [math]\displaystyle{ f : X \to Y }[/math] between two topological spaces proper if the preimage of every compact set in [math]\displaystyle{ Y }[/math] is compact in [math]\displaystyle{ X. }[/math] Other authors call a map [math]\displaystyle{ f }[/math] 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 [math]\displaystyle{ Y }[/math] is compact. The two definitions are equivalent if [math]\displaystyle{ Y }[/math] is locally compact and Hausdorff.

Partial proof of equivalence

Let [math]\displaystyle{ f : X \to Y }[/math] be a closed map, such that [math]\displaystyle{ f^{-1}(y) }[/math] is compact (in [math]\displaystyle{ X }[/math]) for all [math]\displaystyle{ y \in Y. }[/math] Let [math]\displaystyle{ K }[/math] be a compact subset of [math]\displaystyle{ Y. }[/math] It remains to show that [math]\displaystyle{ f^{-1}(K) }[/math] is compact. Let [math]\displaystyle{ \left\{U_a : a \in A\right\} }[/math] be an open cover of [math]\displaystyle{ f^{-1}(K). }[/math] Then for all [math]\displaystyle{ k \in K }[/math] this is also an open cover of [math]\displaystyle{ f^{-1}(k). }[/math] Since the latter is assumed to be compact, it has a finite subcover. In other words, for every [math]\displaystyle{ k \in K, }[/math] there exists a finite subset [math]\displaystyle{ \gamma_k \subseteq A }[/math] such that [math]\displaystyle{ f^{-1}(k) \subseteq \cup_{a \in \gamma_k} U_{a}. }[/math] The set [math]\displaystyle{ X \setminus \cup_{a \in \gamma_k} U_{a} }[/math] is closed in [math]\displaystyle{ X }[/math] and its image under [math]\displaystyle{ f }[/math] is closed in [math]\displaystyle{ Y }[/math] because [math]\displaystyle{ f }[/math] is a closed map. Hence the set [math]\displaystyle{ V_k = Y \setminus f\left(X \setminus \cup_{a \in \gamma_k} U_{a}\right) }[/math] is open in [math]\displaystyle{ Y. }[/math] It follows that [math]\displaystyle{ V_k }[/math] contains the point [math]\displaystyle{ k. }[/math] Now [math]\displaystyle{ K \subseteq \cup_{k \in K} V_k }[/math] and because [math]\displaystyle{ K }[/math] is assumed to be compact, there are finitely many points [math]\displaystyle{ k_1, \dots, k_s }[/math] such that [math]\displaystyle{ K \subseteq \cup_{i =1}^s V_{k_i}. }[/math] Furthermore, the set [math]\displaystyle{ \Gamma = \cup_{i=1}^s \gamma_{k_i} }[/math] is a finite union of finite sets, which makes [math]\displaystyle{ \Gamma }[/math] a finite set. Now it follows that [math]\displaystyle{ f^{-1}(K) \subseteq f^{-1}\left( \cup_{i=1}^s V_{k_i} \right) \subseteq \cup_{a \in \Gamma} U_{a} }[/math] and we have found a finite subcover of [math]\displaystyle{ f^{-1}(K), }[/math] which completes the proof.

If [math]\displaystyle{ X }[/math] is Hausdorff and [math]\displaystyle{ Y }[/math] is locally compact Hausdorff then proper is equivalent to universally closed. A map is universally closed if for any topological space [math]\displaystyle{ Z }[/math] the map [math]\displaystyle{ f \times \operatorname{id}_Z : X \times Z \to Y \times Z }[/math] is closed. In the case that [math]\displaystyle{ Y }[/math] is Hausdorff, this is equivalent to requiring that for any map [math]\displaystyle{ Z \to Y }[/math] the pullback [math]\displaystyle{ X \times_Y Z \to Z }[/math] be closed, as follows from the fact that [math]\displaystyle{ X \times_YZ }[/math] is a closed subspace of [math]\displaystyle{ X \times Z. }[/math]

An equivalent, possibly more intuitive definition when [math]\displaystyle{ X }[/math] and [math]\displaystyle{ Y }[/math] are metric spaces is as follows: we say an infinite sequence of points [math]\displaystyle{ \{p_i\} }[/math] in a topological space [math]\displaystyle{ X }[/math] escapes to infinity if, for every compact set [math]\displaystyle{ S \subseteq X }[/math] only finitely many points [math]\displaystyle{ p_i }[/math] are in [math]\displaystyle{ S. }[/math] Then a continuous map [math]\displaystyle{ f : X \to Y }[/math] is proper if and only if for every sequence of points [math]\displaystyle{ \left\{p_i\right\} }[/math] that escapes to infinity in [math]\displaystyle{ X, }[/math] the sequence [math]\displaystyle{ \left\{f\left(p_i\right)\right\} }[/math] escapes to infinity in [math]\displaystyle{ Y. }[/math]

Properties

• 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 [math]\displaystyle{ f : X \to Y }[/math] is called a compact covering if for every compact subset [math]\displaystyle{ K \subseteq Y }[/math] there exists some compact subset [math]\displaystyle{ C \subseteq X }[/math] such that [math]\displaystyle{ f(C) = K. }[/math]
• A topological space is compact if and only if the map from that space to a single point is proper.
• If [math]\displaystyle{ f : X \to Y }[/math] is a proper continuous map and [math]\displaystyle{ Y }[/math] is a compactly generated Hausdorff space (this includes Hausdorff spaces that are either first-countable or locally compact), then [math]\displaystyle{ f }[/math] is closed.

Generalization

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

See also

• 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

References

• Bourbaki, Nicolas (1998). General topology. Chapters 5–10. Elements of Mathematics. Berlin, New York: Springer-Verlag. ISBN 978-3-540-64563-4. 
• 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. Second series 7 (3): 515-522. doi:10.1112/jlms/s2-7.3.515.