Proper map

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

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.

Definition[edit]

A function f : XY 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 compactly generated 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 × idZ: X × ZY × 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 {pi} in a topological space X escapes to infinity if, for every compact set SX only finitely many points pi are in S. Then a continuous map f : XY 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.

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

Proof of fact[edit]

Let f: X \to 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. 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[edit]

Generalization[edit]

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

See also[edit]

References[edit]

  • 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". Proc. Amer. Math. Soc. 24: 835–836. doi:10.1090/s0002-9939-1970-0254818-x.