Inmathematics, afunction betweentopological spaces is calledproper ifinverse images ofcompact subsets are compact.^[1] Inalgebraic geometry, theanalogous concept is called aproper morphism.

There are several competing definitions of a "properfunction". Some authors call a function${\displaystyle f:X\to Y}$ between twotopological spacesproper if thepreimage of everycompact set in${\displaystyle Y}$ is compact in${\displaystyle X.}$Other authors call a map${\displaystyle f}$proper if it is continuous andclosed with compact fibers; that is if it is acontinuousclosed map and the preimage of every point in${\displaystyle Y}$ iscompact. The two definitions are equivalent if${\displaystyle Y}$ islocally compact andHausdorff.

If${\displaystyle X}$ is Hausdorff and${\displaystyle Y}$ is locally compact Hausdorff then proper is equivalent touniversally closed. A map is universally closed if for any topological space${\displaystyle Z}$ the map${\displaystyle f\times {\mathrm{id}}_Z:X\times Z\to Y\times Z}$ is closed. In the case that${\displaystyle Y}$ is Hausdorff, this is equivalent to requiring that for any map${\displaystyle Z\to Y}$ the pullback${\displaystyle X{\times }_{Y}Z\to Z}$ be closed, as follows from the fact that${\displaystyle X{\times }_{Y}Z}$ is a closed subspace of${\displaystyle X\times Z.}$

An equivalent, possibly more intuitive definition when${\displaystyle X}$ and${\displaystyle Y}$ aremetric spaces is as follows: we say an infinite sequence of points${\displaystyle \{p_i\}}$ in a topological space${\displaystyle X}$escapes to infinity if, for every compact set${\displaystyle S\subseteq X}$ only finitely many points${\displaystyle p_i}$ are in${\displaystyle S.}$ Then a continuous map${\displaystyle f:X\to Y}$ is proper if and only if for every sequence of points${\displaystyle \left\{p_i\right\}}$ that escapes to infinity in${\displaystyle X,}$ the sequence${\displaystyle \left\{f\left(p_i\right)\right\}}$ escapes to infinity in${\displaystyle Y.}$

• Every continuous map from a compact space to aHausdorff space is both proper andclosed.
• Everysurjective proper map is a compact covering map.
  • A map${\displaystyle f:X\to Y}$ is called acompact covering if for every compact subset${\displaystyle K\subseteq Y}$ there exists some compact subset${\displaystyle C\subseteq X}$ such that${\displaystyle f(C)=K.}$
• A topological space is compact if and only if the map from that space to a single point is proper.
• If${\displaystyle f:X\to Y}$ is a proper continuous map and${\displaystyle Y}$ is acompactly generated Hausdorff space (this includes Hausdorff spaces that are eitherfirst-countable orlocally compact), then${\displaystyle f}$ is closed.^[2]

It is possible to generalize the notion of proper maps of topological spaces tolocales andtopoi, see (Johnstone 2002).

• 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

• Theory of continuous functions

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