Proper map explained

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

f : X \to Y between two topological spaces if the preimage of every compact set in Y is compact in X. Other authors call a map f if it is continuous and ; that is if it is a continuous closed map and the preimage of every point in Y is compact. The two definitions are equivalent if Y is locally compact and Hausdorff.Let f : X \to Y be a closed map, such that f ^{-1} (y) is compact (in X ) for all y \in Y. Let K be a compact subset of Y. It remains to show that f ^{-1} (K) is compact.

Let

\left\{U _{a} : a \in A\right\} be an open cover of f ^{-1} (K). Then for all k \in K this is also an open cover of f ^{-1} (k). Since the latter is assumed to be compact, it has a finite subcover. In other words, for every k \in K, there exists a finite subset \gamma _{k} \subseteq A such that f ^{-1} (k) \subseteq \cup _{a \in \gamma _{k}} U _{a} . The set X \setminus \cup _{a \in \gamma _{k}} U _{a} is closed in X and its image under f is closed in Y because f is a closed map. Hence the set$$V\_k\; =\; Y\; \backslash setminus\; f\backslash left(X\; \backslash setminus\; \backslash cup\_\; U\_\backslash right)$$ is open in Y. It follows that V _{k} contains the point k. Now K \subseteq \cup _{k} V _{k} and because K is assumed to be compact, there are finitely many points k _{1,} ... , k _{s} such that K \subseteq \cup ^{s} _{i =1} V _{k _{i}} . Furthermore, the set \Gamma = \cup ^{s} _{i =1} \gamma _{k _{i}} is a finite union of finite sets, which makes \Gamma a finite set.

Now it follows that

f ^{-1} (K) \subseteq f ^{-1} \left( \cup ^{s} _{i =1} V _{k _{i}} \right) \subseteq \cup _{a} U _{a} and we have found a finite subcover of f ^{-1} (K), which completes the proof.

If

X is Hausdorff and Y is locally compact Hausdorff then proper is equivalent to . A map is universally closed if for any topological space Z the map f x \operatorname{id} _{Z} : X x Z \to Y x Z is closed. In the case that Y is Hausdorff, this is equivalent to requiring that for any map Z \to Y the pullback X x _{Y} Z \to Z be closed, as follows from the fact that X x _{YZ} is a closed subspace of X x Z.

An equivalent, possibly more intuitive definition when

X and Y are metric spaces is as follows: we say an infinite sequence of points \{p _{i\}} in a topological space X if, for every compact set S \subseteq X only finitely many points p _{i} are in S. Then a continuous map f : X \to Y is proper if and only if for every sequence of points \left\{p _{i\right\}} that escapes to infinity in X, the sequence \left\{f\left(p _{i\right)\right\}} escapes to infinity in Y.

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

f : X \to Y is called a if for every compact subset K \subseteq Y there exists some compact subset C \subseteq X such that f(C) = K.

• A topological space is compact if and only if the map from that space to a single point is proper.
• If

f : X \to Y is a proper continuous map and Y is a compactly generated Hausdorff space (this includes Hausdorff spaces that are either first-countable or locally compact), then f is closed.^[1]

Generalization

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

References

• Book: Bourbaki . Nicolas . Nicolas Bourbaki . General topology. Chapters 5–10 . . Berlin, New York . Elements of Mathematics . 978-3-540-64563-4 . 1726872 . 1998.
• Book: Johnstone, Peter . Peter Johnstone (mathematician). Sketches of an elephant: a topos theory compendium . . Oxford . 2002 . 0-19-851598-7 ., esp. section C3.2 "Proper maps"
• Book: Brown, Ronald . Ronald Brown (mathematician). Topology and groupoids . . North Carolina . 2006 . 1-4196-2722-8 ., esp. p. 90 "Proper maps" and the Exercises to Section 3.6.
• Brown . Ronald . Ronald Brown (mathematician). Sequentially proper maps and a sequential compactification. . Second series. 7 . 3 . 1973. 515-522 . 10.1112/jlms/s2-7.3.515.

Notes and References

1. Palais. Richard S.. Richard Palais. When proper maps are closed. Proceedings of the American Mathematical Society. 1970. 24. 4. 835–836. 10.1090/s0002-9939-1970-0254818-x. free. 0254818 .