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 : X → 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 pre-image of every point in Y is compact. The two definitions are equivalent if Y is locally compact and Hausdorff. For a proof of this fact see the end of this section. More abstractly, f is proper if f is universally closed, i.e. if for any topological space Z the map

f × id_Z: X × Z → Y × Z

is closed. These definitions are 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 ⊂ X only finitely many points p_i are in S. Then a continuous map f : X → Y is proper if and only if for every sequence of points {p_i} that escapes to infinity in X, {f(p_i)} escapes to infinity in Y.

This last sequential idea looks like being related to the notion of sequentially proper, see a reference below.

Proof of fact

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 $Y$. 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.

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 : X → 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).

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, N. Carolina: Booksurge, ISBN 1-4196-2722-8, esp. p. 90 "Proper maps" and the Exercises to Section 3.6.
• Brown, R. "Sequentially proper maps and a sequential compactification", J. London Math Soc. (2) 7 (1973) 515-522.

1. Palais, Richard S. (1970). "When proper maps are closed" (PDF). Proc. Amer. Math. Soc. 24: 835–836. doi:10.1090/s0002-9939-1970-0254818-x.