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 $f:X\rightarrow Y$ is a morphism of algebraic varieties, then $f(X)$ (and, moreover, the image under $f$ of any constructible subset in $X$) is a constructible subset in $Y$. This is related to the fact that "algebraic" conditions determine the constructible subsets of an algebraic variety.

A mapping $h:X\rightarrow T$ is called constructible if $h(X)$ is finite and if for any point $t\in T$ the pre-image $h^{-1}(t)$ is a constructible subset in $X$.

References

[1]	A. Grothendieck, J. Dieudonné, "Eléments de géometrie algébrique" , I. Le langage des schémes , Springer (1971) MR0217085 Zbl 0203.23301
[2]	A. Borel, "Linear algebraic groups" , Benjamin (1969) MR0251042 Zbl 0206.49801 Zbl 0186.33201