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 → Y {\displaystyle f:X\to Y} between two topological spaces proper if the preimage of every compact set in Y {\displaystyle Y} is compact in X . {\displaystyle X.}
Other authors call a map f {\displaystyle f} proper if it is continuous and closed with compact fibers; that is if it is a continuous closed map and the preimage of every point in Y {\displaystyle Y} is compact. The two definitions are equivalent if Y {\displaystyle Y} is locally compact and Hausdorff.
If X {\displaystyle X} is Hausdorff and Y {\displaystyle Y} is locally compact Hausdorff then proper is equivalent to universally closed. A map is universally closed if for any topological space Z {\displaystyle Z} the map f × id Z : X × Z → Y × Z {\displaystyle f\times \operatorname {id} _{Z}:X\times Z\to Y\times Z} is closed. In the case that Y {\displaystyle Y} is Hausdorff, this is equivalent to requiring that for any map Z → Y {\displaystyle Z\to Y} the pullback X × Y Z → Z {\displaystyle X\times _{Y}Z\to Z} be closed, as follows from the fact that X × Y Z {\displaystyle X\times _{Y}Z} is a closed subspace of X × Z . {\displaystyle X\times Z.}
An equivalent, possibly more intuitive definition when X {\displaystyle X} and Y {\displaystyle Y} are metric spaces is as follows: we say an infinite sequence of points { p i } {\displaystyle \{p_{i}\}} in a topological space X {\displaystyle X} escapes to infinity if, for every compact set S ⊆ X {\displaystyle S\subseteq X} only finitely many points p i {\displaystyle p_{i}} are in S . {\displaystyle S.} Then a continuous map f : X → Y {\displaystyle f:X\to Y} is proper if and only if for every sequence of points { p i } {\displaystyle \left\{p_{i}\right\}} that escapes to infinity in X , {\displaystyle X,} the sequence { f ( p i ) } {\displaystyle \left\{f\left(p_{i}\right)\right\}} escapes to infinity in Y . {\displaystyle 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 → Y {\displaystyle f:X\to Y} is called a compact covering if for every compact subset K ⊆ Y {\displaystyle K\subseteq Y} there exists some compact subset C ⊆ X {\displaystyle C\subseteq X} such that f ( C ) = K . {\displaystyle 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 → Y {\displaystyle f:X\to Y} is a proper continuous map and Y {\displaystyle Y} is a compactly generated Hausdorff space (this includes Hausdorff spaces that are either first-countable or locally compact), then f {\displaystyle f} is closed.
Generalization It is possible to generalize the notion of proper maps of topological spaces to locales and topoi, see (Johnstone 2002).
See also Almost open map – Map that satisfies a condition similar to that of being an open map Open and closed maps – Functions that send open (resp. closed) subsets to open (resp. closed) subsets Perfect map – Continuous closed surjective map, each of whose fibers are also compact sets Topology glossary
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