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:''For a [[Gödel constructive set]], see [[constructible universe]].''
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In [[topology]], a '''constructible set''' in a [[topological space]] is a finite union of [[locally closed set]]s. (A set is locally closed if it is the intersection of an open set and closed set, or equivalently, if it is open in its closure.) Constructible sets form a [[Boolean algebra (structure)|Boolean algebra]] (i.e., it is closed under finite union and complementation.) In fact, the constructible sets are precisely the Boolean algebra generated by open sets and closed sets; hence, the name "constructible". The notion appears in classical [[algebraic geometry]].


Chevalley's theorem (EGA IV, 1.8.4.) states: Let <math>f: X \to Y</math> be a morphism of finite presentation of schemes. Then the image of any constructible set under ''f'' is constructible. In particular, the image of a variety need not be a variety, but is (under the assumptions) always a constructible set. For example, the projection of the variety <math>xy=1</math> (the hyperbola) onto the ''x''-axis is the ''x''-axis minus the origin: this is a constructible set which is neither a variety nor open or closed in the plane.
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In a topological space, every constructible set contains a dense open subset of its closure.<ref>Jinpeng An (2012). [http://www.springerlink.com/content/73hg840753675717/ "Rigid geometric structures, isometric actions, and algebraic quotients"]. Geom. Dedicata '''157''': 153–185.</ref>
 
== See also ==
*[[Constructible topology]]
*[[Constructible sheaf]]
 
==Notes==
{{reflist}}
 
== References ==
* Allouche, Jean Paul. ''[http://www.lri.fr/~allouche/allouche96e.ps Note on the constructible sets of a topological space].''
* {{cite book|last1=Andradas|first1=Carlos|last2=Bröcker|first2=Ludwig|last3=Ruiz|first3=Jesús&nbsp;M.|title=Constructible sets in real geometry | series=Ergebnisse der Mathematik und ihrer Grenzgebiete (3) --- Results in Mathematics and Related Areas (3)| volume=33 |  publisher=Springer-Verlag | location=Berlin | year=1996 | pages=x+270 | isbn=3-540-60451-0 | mr=1393194 }}
* [[Armand Borel|Borel, Armand]]. ''Linear algebraic groups.''
* [[Alexander Grothendieck|Grothendieck, Alexander]]. ''EGA 0 §9''
*{{EGA|book=1-2}}
*{{EGA|book=1| pages = 5–228}}
* {{ cite book|last=Mostowski|first=A.| authorlink=Andrzej Mostowski | title=Constructible sets with applications | series=Studies in Logic and the Foundations of Mathematics | publisher=North-Holland Publishing Co. ---- [[Polish Scientific Publishers|PWN-Polish Scientific Publishers]] | location=Amsterdam --- Warsaw| year=1969 | pages=ix+269 | mr=255390 }}
 
[[Category:Topology]]
[[Category:Algebraic geometry]]
 
 
{{topology-stub}}

Revision as of 23:24, 26 February 2014

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