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:''For a [[Gödel constructive set]], see [[constructible universe]].''
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 20:27, 26 December 2013

For a Gödel constructive set, see constructible universe.

In topology, a constructible set in a topological space is a finite union of locally closed sets. (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 (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 f:X→Y 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 xy=1 (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.

In a topological space, every constructible set contains a dense open subset of its closure.[1]

See also

Notes

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References


Template:Topology-stub

  1. ↑ Jinpeng An (2012). "Rigid geometric structures, isometric actions, and algebraic quotients". Geom. Dedicata 157: 153–185.
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