Proper map

Definition

There are several competing definitions of a "proper function". Some authors call a function f : X \to Y between two topological spaces proper if the preimage of every compact set in Y is compact in X. Other authors call a map f 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 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 \setminus f\left(X \setminus \cup_{a \in \gamma_k} U_{a}\right) is open in Y. It follows that V_k contains the point k. Now K \subseteq \cup_{k \in K} V_k and because K is assumed to be compact, there are finitely many points k_1, \dots, k_s such that K \subseteq \cup_{i =1}^s V_{k_i}. Furthermore, the set \Gamma = \cup_{i=1}^s \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_{i=1}^s V_{k_i} \right) \subseteq \cup_{a \in \Gamma} 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 universally closed. A map is universally closed if for any topological space Z the map f \times \operatorname{id}_Z : X \times Z \to Y \times 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 \times_Y Z \to Z be closed, as follows from the fact that X \times_YZ is a closed subspace of X \times 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 escapes to infinity 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 compact covering 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.

Generalization

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

Citations

References

• , esp. section C3.2 "Proper maps"
• , esp. p. 90 "Proper maps" and the Exercises to Section 3.6.

References

1. Palais, Richard S.. (1970). "When proper maps are closed". [[Proceedings of the American Mathematical Society]].