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.

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== There are several competing definitions of a "proper function". Some authors call a function f : X \to Y between two topological spaces if the preimage of every compact set in Y is compact in X. Other authors call a map f if it is continuous and ; that is if it is a continuous closed map and the preimage of every point in Y is compact. The two definitions are equivalent if Y is locally compact and Hausdorff. 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. It remains to show that f^{-1}(K) is compact. Let \left\{U_a : a \in A\right\} 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 every k \in K, there exists a finite subset \gamma_k \subseteq A such that f^{-1}(k) \subseteq \cup_{a \in \gamma_k} U_{a}. The set X \setminus \cup_{a \in \gamma_k} U_{a} is closed in X and its image under f is closed in Y because f is a closed map. Hence the set V_k = Y \setminus f\left(X \setminus \cup_{a \in \gamma_k} U_{a}\right) is open in Y. It follows that V_k contains the point k. Now K \subseteq \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 \subseteq \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, which makes \Gamma a finite set. Now it follows that f^{-1}(K) \subseteq f^{-1}\left( \cup_{i=1}^s V_{k_i} \right) \subseteq \cup_{a \in \Gamma} U_{a} 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 . A map is universally closed if for any topological space Z the map f \times \operatorname{id}_Z : 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_YZ 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 if, for every compact set S \subseteq X only finitely many points p_i are in S. Then a continuous map f : X \to Y is proper if and only if for every sequence of points \left\{p_i\right\} that escapes to infinity in X, the sequence \left\{f\left(p_i\right)\right\} escapes to infinity in Y. ==Properties== Every continuous map from a compact space to a Hausdorff space is both proper and closed. Every surjective proper map is a compact covering map. A map f : X \to Y is called a if for every compact subset K \subseteq Y there exists some compact subset C \subseteq X such that f(C) = K. A topological space is compact if and only if the map from that space to a single point is proper. If f : X \to Y is a proper continuous map and Y is a compactly generated Hausdorff space (this includes Hausdorff spaces that 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 . ==See also== ==Citations== ==References== , esp. section C3.2 "Proper maps" , esp. p. 90 "Proper maps" and the Exercises to Section 3.6.

Text derived from https://en.wikipedia.org/w/index.php?oldid=1328730394 under CC BY-SA 4.0; modifications have been made.