# Constructible subset

*of an algebraic variety*

A finite union of locally closed (in the Zariski topology) subsets. A locally closed subset is, by definition, an intersection of an open and a closed subset. The constructible subsets form a Boolean algebra and can be defined as elements of the Boolean algebra generated by the algebraic subvarieties. The role of constructible subsets in algebraic geometry is revealed by Chevalley's theorem: If is a morphism of algebraic varieties, then (and, moreover, the image of any constructible subset in ) is a constructible subset in . This is related to the fact that "algebraic" conditions determine the constructible subsets of an algebraic variety.

A mapping is called constructible if is finite and if for any point the pre-image is a constructible subset in .

#### References

[1] | A. Grothendieck, J. Dieudonné, "Eléments de géometrie algébrique" , I. Le langage des schémes , Springer (1971) |

[2] | A. Borel, "Linear algebraic groups" , Benjamin (1969) |

**How to Cite This Entry:**

Constructible subset.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Constructible_subset&oldid=15045