Proper map

This article is about the concept in topology. For the concept in convex analysis, see proper convex function.

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 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.

More abstractly, f is proper if f is universally closed, i.e., if for any topological space Z the map

$f\times \operatorname {id} _{Z}\colon X\times Z\to Y\times Z$

is closed. This definition is equivalent to the previous one if X is Hausdorff and Y is locally compact Hausdorff.

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.

Generalization

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

See also

• Perfect map
• Topology glossary

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.