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 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 such that The set is closed in X and its image under f is closed in Y because f is a closed map. Hence the set is open in Y. It follows that V_k contains the point k. Now and because K is assumed to be compact, there are finitely many points such that Furthermore, the set is a finite union of finite sets, which makes \Gamma a finite set. Now it follows that 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 is closed. In the case that Y is Hausdorff, this is equivalent to requiring that for any map Z \to Y the pullback 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 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**** **** that**** escapes to**** infinity**** in**** X,**** the sequence**** **** escapes to**** infinity**** in**** Y.****

Properties

Generalization

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

Citations