Proper map

Map between topological spaces with the property that the preimage of every compact is compact

In mathematics, a function between topological spaces is called proper if inverse images of compact subsets are compact.^[1] 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 ${\displaystyle f:X\to Y}$ between two topological spaces proper if the preimage of every compact set in ${\displaystyle Y}$ is compact in ${\displaystyle X.}$Other authors call a map ${\displaystyle f}$ 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 ${\displaystyle Y}$ is compact. The two definitions are equivalent if ${\displaystyle Y}$ is locally compact and Hausdorff.

If ${\displaystyle X}$ is Hausdorff and ${\displaystyle Y}$ is locally compact Hausdorff then proper is equivalent to universally closed. A map is universally closed if for any topological space ${\displaystyle Z}$ the map ${\displaystyle f\times \operatorname {id} _{Z}:X\times Z\to Y\times Z}$ 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 ${\displaystyle Y}$ are metric 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 ${\displaystyle Y.}$

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 ${\displaystyle f:X\to Y}$ is called a compact 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 a compactly generated Hausdorff space (this includes Hausdorff spaces that are either first-countable or locally compact), then ${\displaystyle f}$ is closed.^[2]

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 – A function that sends 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 – Mathematics glossaryPages displaying short descriptions of redirect targets

Citations

1. Lee 2012, p. 610, above Prop. A.53.
2. Palais, Richard S. (1970). "When proper maps are closed". Proceedings of the American Mathematical Society. 24 (4): 835–836. doi:10.1090/s0002-9939-1970-0254818-x. MR 0254818.

References

• Bourbaki, Nicolas (1998). General topology. Chapters 5–10. Elements of Mathematics. Berlin, New York: Springer-Verlag. ISBN 978-3-540-64563-4. MR 1726872.
• Johnstone, Peter (2002). Sketches of an elephant: a topos theory compendium. Oxford: Oxford University Press. ISBN 0-19-851598-7., esp. section C3.2 "Proper maps"
• Brown, Ronald (2006). Topology and groupoids. North Carolina: Booksurge. ISBN 1-4196-2722-8., esp. p. 90 "Proper maps" and the Exercises to Section 3.6.
• Brown, Ronald (1973). "Sequentially proper maps and a sequential compactification". Journal of the London Mathematical Society. Second series. 7 (3): 515–522. doi:10.1112/jlms/s2-7.3.515.
• Lee, John M. (2012). Introduction to Smooth Manifolds. Graduate Texts in Mathematics. Vol. 218 (Second ed.). New York London: Springer-Verlag. ISBN 978-1-4419-9981-8. OCLC 808682771.