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| {{About|the concept in [[topology]]|the concept in [[convex analysis]]|proper convex function}}
| | Nice to satisfy you, I am Marvella Shryock. One of the issues she enjoys most is to read comics and she'll be beginning something else alongside with it. In her professional life she is a payroll clerk but she's always wanted her personal company. Her family members lives in Minnesota.<br><br>Here is my blog; home std test, [http://rtdcs.hufs.ac.kr/?document_srl=780411 visit the website], |
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| In [[mathematics]], a [[continuous function]] between [[topological space]]s is called '''proper''' if [[inverse image]]s of [[compact space|compact subsets]] are compact. In [[algebraic geometry]], the analogous concept is called a [[proper morphism]].
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| == Definition ==
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| A [[function (mathematics)|function]] ''f'' : ''X'' → ''Y'' between two [[topological space]]s is '''proper''' if the [[preimage]] of every [[compact space|compact]] set in ''Y'' is compact in ''X''.
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| There are several competing descriptions. For instance, a continuous map ''f'' is proper if it is a [[closed map]] and the pre-image of every point in ''Y'' is compact. The two definitions are equivalent if Y is compactly generated and Hausdorff. For a proof of this fact see the end of this section. More abstractly, ''f'' is proper if ''f'' is universally closed, i.e. if for any topological space ''Z'' the map
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| :''f'' × id<sub>''Z''</sub>: ''X'' × ''Z'' → ''Y'' × ''Z''
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| is closed. These definitions are equivalent to the previous one if ''X'' is [[Hausdorff space|Hausdorff]] and ''Y'' is [[Locally compact space|locally compact]] Hausdorff.
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| An equivalent, possibly more intuitive definition when ''X'' and ''Y'' are [[metric space]]s is as follows: we say an infinite sequence of points {''p''<sub>''i''</sub>} in a topological space ''X'' '''escapes to infinity''' if, for every compact set ''S'' ⊂ ''X'' only finitely many points ''p''<sub>''i''</sub> are in ''S''. Then a continuous map ''f'' : ''X'' → ''Y'' is proper if and only if for every sequence of points {''p''<sub>''i''</sub>} that escapes to infinity in ''X'', {''f''(''p''<sub>''i''</sub>)} escapes to infinity in ''Y''.
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| This last sequential idea looks like being related to the notion of sequentially proper, see a reference below.
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| === Proof of fact ===
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| Let <math>f: X \to Y</math> be a continuous closed map, such that <math>f^{-1}(y)</math> is compact (in X) for all <math>y \in Y</math>. Let <math>K</math> be a compact subset of <math>Y</math>. We will show that <math>f^{-1}(K)</math> is compact.
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| Let <math>\{ U_{\lambda} \vert \lambda\ \in\ \Lambda \}</math> be an open cover of <math>f^{-1}(K)</math>. Then for all <math>k\ \in K</math> this is also an open cover of <math>f^{-1}(k)</math>. Since the latter is assumed to be compact, it has a finite subcover. In other words, for all <math>k\ \in K</math> there is a finite set <math>\gamma_k \subset \Lambda</math> such that <math>f^{-1}(k) \subset \cup_{\lambda \in \gamma_k} U_{\lambda}</math>.
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| The set <math>X \setminus \cup_{\lambda \in \gamma_k} U_{\lambda}</math> is closed. Its image is closed in Y, because f is a closed map. Hence the set
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| <math>V_k = Y \setminus f(X \setminus \cup_{\lambda \in \gamma_k} U_{\lambda})</math> is open in Y. It is easy to check that <math>V_k</math> contains the point <math>k</math>. | |
| Now <math>K \subset \cup_{k \in K} V_k</math> and because K is assumed to be compact, there are finitely many points <math>k_1,\dots , k_s</math> such that <math>K \subset \cup_{i =1}^s V_{k_i}</math>. Furthermore the set <math>\Gamma = \cup_{i =1}^s \gamma_{k_i} </math> is a finite union of finite sets, thus <math>\Gamma</math> is finite.
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| Now it follows that <math>f^{-1}(K) \subset f^{-1}(\cup_{i=1}^s V_{k_i}) \subset \cup_{\lambda \in \Gamma} U_{\lambda}</math> and we have found a finite subcover of <math>f^{-1}(K)</math>, which completes the proof.
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| == Properties ==
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| *A topological space is compact if and only if the map from that space to a single point is proper.
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| *Every continuous map from a compact space to a [[Hausdorff space]] is both proper and [[closed map|closed]].
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| *If ''f'' : ''X'' → ''Y'' is a proper continuous map and ''Y'' is a [[compactly generated Hausdorff space]] (this includes Hausdorff spaces which are either [[first-countable]] or [[locally compact]]), then ''f'' is closed.<ref name=palais>{{cite journal|last=Palais|first=Richard S.|title=When proper maps are closed|journal=Proc. Amer. Math. Soc.|year=1970|volume=24|pages=835–836|url=http://www.ams.org/journals/proc/1970-024-04/S0002-9939-1970-0254818-X/S0002-9939-1970-0254818-X.pdf}}</ref>
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| == Generalization ==
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| It is possible to generalize
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| the notion of proper maps of topological spaces to [[Pointless topology|locales]] and [[topos|topoi]], see {{Harv|Johnstone|2002}}. | |
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| == See also ==
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| * [[Perfect map]]
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| * [[Topology glossary]]
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| ==References==
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| * {{Citation | last1=Bourbaki | first1=Nicolas | author1-link = Nicolas Bourbaki | title=General topology. Chapters 5--10 | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Elements of Mathematics | isbn=978-3-540-64563-4 | id={{MathSciNet | id = 1726872}} | year=1998}}
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| * {{citation |last=Johnstone |first=Peter |title=Sketches of an elephant: a topos theory compendium |publisher=[[Oxford University Press]] |location=Oxford |year=2002 |pages= |isbn=0-19-851598-7 |oclc= |doi=}}, esp. section C3.2 "Proper maps"
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| * {{citation |last=Brown |first=Ronald |title=Topology and groupoids |publisher=[[Booksurge]] |location= N. Carolina |year=2006 |pages= |isbn=1-4196-2722-8 |oclc= |doi=}}, esp. p. 90 "Proper maps" and the Exercises to Section 3.6.
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| * Brown, R. "Sequentially proper maps and a sequential compactification", J. London Math Soc. (2) 7 (1973) 515-522.
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| {{Reflist}}
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| [[Category:Continuous mappings]]
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Nice to satisfy you, I am Marvella Shryock. One of the issues she enjoys most is to read comics and she'll be beginning something else alongside with it. In her professional life she is a payroll clerk but she's always wanted her personal company. Her family members lives in Minnesota.
Here is my blog; home std test, visit the website,