Proper map

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

A function $f\colon X\to Y$ between two topological spaces is proper if the preimage of every compact set in Y is compact in X.

There are several competing descriptions. For instance, a continuous map f is proper if it is closed with compact fibers, i.e. if it is a closed map and the preimage of every point in Y is compact. The two definitions are equivalent if Y is locally compact and Hausdorff.

Proof of equivalence

Let $f\colon X\to Y$ 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 $Y$ . We will show that $f^{-1}(K)$ is compact.

Let $\{ U_{\lambda} \vert \lambda\ \in\ \Lambda \}$ 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 all $k\ \in K$ there is a finite set $\gamma_k \subset \Lambda$ such that $f^{-1}(k) \subset \cup_{\lambda \in \gamma_k} U_{\lambda}$ . The set $X \setminus \cup_{\lambda \in \gamma_k} U_{\lambda}$ is closed. Its image is closed in Y, because f is a closed map. Hence the set

$V_k = Y \setminus f(X \setminus \cup_{\lambda \in \gamma_k} U_{\lambda})$ is open in Y. It is easy to check that $V_{k}$ contains the point $k$ . Now $K \subset \cup_{k \in K} V_k$ and because K is assumed to be compact, there are finitely many points $k_1,\dots , k_s$ such that $K \subset \cup_{i =1}^s V_{k_i}$ . Furthermore the set $\Gamma = \cup_{i =1}^s \gamma_{k_i}$ is a finite union of finite sets, thus $\Gamma$ is finite.

Now it follows that $f^{-1}(K) \subset f^{-1}(\cup_{i=1}^s V_{k_i}) \subset \cup_{\lambda \in \Gamma} U_{\lambda}$ 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 $f\times \operatorname {id} _{Z}\colon X\times Z\to Y\times Z$ 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 $X\times _{Y}Z$ 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 $S\subset X$ only finitely many points $p_{i}$ are in S. Then a continuous map $f\colon X\to Y$ is proper if and only if for every sequence of points $\{p_i\}$ that escapes to infinity in X, the sequence $\{f(p_{i})\}$ escapes to infinity in Y.

Properties

A topological space is compact if and only if the map from that space to a single point is proper.
Every continuous map from a compact space to a Hausdorff space is both proper and closed.
If $f\colon X\to Y$ is a proper continuous map and Y is a compactly generated Hausdorff space (this includes Hausdorff spaces which 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 (Johnstone 2002).

References

Bourbaki, Nicolas (1998). General topology. Chapters 5--10. Elements of Mathematics. Berlin, New York: Springer-Verlag. ISBN 978-3-540-64563-4. MR 1726872.
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. 2. 7: 515–522.

↑ Palais, Richard S. (1970). "When proper maps are closed" (PDF). Proceedings of the American Mathematical Society. 24: 835–836. doi:10.1090/s0002-9939-1970-0254818-x.

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[palais-1] Palais, Richard S. (1970). "When proper maps are closed" (PDF). Proceedings of the American Mathematical Society. 24: 835–836. doi:10.1090/s0002-9939-1970-0254818-x.

Proper map

Definition

Properties

Generalization

See also

References