There are several competing definitions of a "proper function". Some authors call a function between two topological spacesproper if the preimage of every compact set in is compact in Other authors call a map proper if it is continuous and closed with compact fibers; that is if it is a continuousclosed map and the preimage of every point in is compact. The two definitions are equivalent if is locally compact and Hausdorff.
Partial proof of equivalence
Let be a closed map, such that is compact (in ) for all Let be a compact subset of It remains to show that is compact.
Let be an open cover of Then for all this is also an open cover of Since the latter is assumed to be compact, it has a finite subcover. In other words, for every there exists a finite subset such that The set is closed in and its image under is closed in because is a closed map. Hence the set
is open in It follows that contains the point Now and because is assumed to be compact, there are finitely many points such that Furthermore, the set is a finite union of finite sets, which makes a finite set.
Now it follows that and we have found a finite subcover of which completes the proof.
If is Hausdorff and is locally compact Hausdorff then proper is equivalent to universally closed. A map is universally closed if for any topological space the map is closed. In the case that is Hausdorff, this is equivalent to requiring that for any map the pullback be closed, as follows from the fact that is a closed subspace of
An equivalent, possibly more intuitive definition when and are metric spaces is as follows: we say an infinite sequence of points in a topological space escapes to infinity if, for every compact set only finitely many points are in Then a continuous map is proper if and only if for every sequence of points that escapes to infinity in the sequence escapes to infinity in
Propertiesedit
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 is called a compact covering if for every compact subset there exists some compact subset such that
A topological space is compact if and only if the map from that space to a single point is proper.
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This article is about the concept in topology For the concept in convex analysis see proper convex function 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 Contents 1 Definition 2 Properties 3 Generalization 4 See also 5 Citations 6 ReferencesDefinition editThere are several competing definitions of a proper function Some authors call a function f X Y displaystyle f X to Y nbsp between two topological spaces proper if the preimage of every compact set in Y displaystyle Y nbsp is compact in X displaystyle X nbsp Other authors call a map f displaystyle f nbsp 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 nbsp is compact The two definitions are equivalent if Y displaystyle Y nbsp is locally compact and Hausdorff Partial proof of equivalenceLet f X Y displaystyle f X to Y nbsp be a closed map such that f 1 y displaystyle f 1 y nbsp is compact in X displaystyle X nbsp for all y Y displaystyle y in Y nbsp Let K displaystyle K nbsp be a compact subset of Y displaystyle Y nbsp It remains to show that f 1 K displaystyle f 1 K nbsp is compact Let U a a A displaystyle left U a a in A right nbsp be an open cover of f 1 K displaystyle f 1 K nbsp Then for all k K displaystyle k in K nbsp this is also an open cover of f 1 k displaystyle f 1 k nbsp Since the latter is assumed to be compact it has a finite subcover In other words for every k K displaystyle k in K nbsp there exists a finite subset g k A displaystyle gamma k subseteq A nbsp such that f 1 k a g k U a displaystyle f 1 k subseteq cup a in gamma k U a nbsp The set X a g k U a displaystyle X setminus cup a in gamma k U a nbsp is closed in X displaystyle X nbsp and its image under f displaystyle f nbsp is closed in Y displaystyle Y nbsp because f displaystyle f nbsp is a closed map Hence the setV k Y f X a g k U a displaystyle V k Y setminus f left X setminus cup a in gamma k U a right nbsp is open in Y displaystyle Y nbsp It follows that V k displaystyle V k nbsp contains the point k displaystyle k nbsp Now K k K V k displaystyle K subseteq cup k in K V k nbsp and because K displaystyle K nbsp is assumed to be compact there are finitely many points k 1 k s displaystyle k 1 dots k s nbsp such that K i 1 s V k i displaystyle K subseteq cup i 1 s V k i nbsp Furthermore the set G i 1 s g k i displaystyle Gamma cup i 1 s gamma k i nbsp is a finite union of finite sets which makes G displaystyle Gamma nbsp a finite set Now it follows that f 1 K f 1 i 1 s V k i a G U a displaystyle f 1 K subseteq f 1 left cup i 1 s V k i right subseteq cup a in Gamma U a nbsp and we have found a finite subcover of f 1 K displaystyle f 1 K nbsp which completes the proof If X displaystyle X nbsp is Hausdorff and Y displaystyle Y nbsp is locally compact Hausdorff then proper is equivalent to universally closed A map is universally closed if for any topological space Z displaystyle Z nbsp the map f id Z X Z Y Z displaystyle f times operatorname id Z X times Z to Y times Z nbsp is closed In the case that Y displaystyle Y nbsp is Hausdorff this is equivalent to requiring that for any map Z Y displaystyle Z to Y nbsp the pullback X Y Z Z displaystyle X times Y Z to Z nbsp be closed as follows from the fact that X Y Z displaystyle X times Y Z nbsp is a closed subspace of X Z displaystyle X times Z nbsp An equivalent possibly more intuitive definition when X displaystyle X nbsp and Y displaystyle Y nbsp are metric spaces is as follows we say an infinite sequence of points p i displaystyle p i nbsp in a topological space X displaystyle X nbsp escapes to infinity if for every compact set S X displaystyle S subseteq X nbsp only finitely many points p i displaystyle p i nbsp are in S displaystyle S nbsp Then a continuous map f X Y displaystyle f X to Y nbsp is proper if and only if for every sequence of points p i displaystyle left p i right nbsp that escapes to infinity in X displaystyle X nbsp the sequence f p i displaystyle left f left p i right right nbsp escapes to infinity in Y displaystyle Y nbsp Properties editEvery 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 nbsp is called a compact covering if for every compact subset K Y displaystyle K subseteq Y nbsp there exists some compact subset C X displaystyle C subseteq X nbsp such that f C K displaystyle f C K nbsp 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 nbsp is a proper continuous map and Y displaystyle Y nbsp is a compactly generated Hausdorff space this includes Hausdorff spaces that are either first countable or locally compact then f displaystyle f nbsp is closed 2 Generalization editIt is possible to generalize the notion of proper maps of topological spaces to locales and topoi see Johnstone 2002 See also editAlmost 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 targetsCitations edit Lee 2012 p 610 above Prop A 53 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 editBourbaki 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 Retrieved from https en wikipedia org w index php title Proper map amp oldid 1154139256, wikipedia, wiki, book, books, library,