Proper map

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In mathematics, a continuous 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.

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[edit] Definition

A function f : X → Y between two topological spaces is proper if and only 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. For a proof of this fact see the end of this section. More abstractly, f is proper if it is a closed map, and for any space Z the map

(f, idZ): X × Z → Y × Z

is closed. These definitions are equivalent to the previous one if the space X is locally compact.

An equivalent, possibly more intuitive definition is as follows: we say an infinite sequence of points {pi} in a topological space X escapes to infinity if, for every compact set S ⊂ X only finitely many points pi are in S. Then a map f : X → Y is proper if and only if for every sequence of points {pi} that escapes to infinity in X, {f(pi)} escapes to infinity in Y.

[edit] Proof of fact

(someone with good wiki-editing skills should make the math look nicer) Let f: X->Y be a continuous 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. It's 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 Vk 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 Γ 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), wich completes the proof.

[edit] Properties

[edit] Generalization

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

[edit] See also

[edit] References