View source for Constructible subset

''of an algebraic variety''

A finite union of locally closed (in the [[Zariski topology|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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025300/c0253001.png" /> is a morphism of algebraic varieties, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025300/c0253002.png" /> (and, moreover, the image of any constructible subset in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025300/c0253003.png" />) is a constructible subset in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025300/c0253004.png" />. This is related to the fact that  "algebraic"  conditions determine the constructible subsets of an algebraic variety.

A mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025300/c0253005.png" /> is called constructible if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025300/c0253006.png" /> is finite and if for any point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025300/c0253007.png" /> the pre-image <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025300/c0253008.png" /> is a constructible subset in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025300/c0253009.png" />.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A. Grothendieck,   J. Dieudonné,   "Eléments de géometrie algébrique" , '''I. Le langage des schémes''' , Springer  (1971)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A. Borel,   "Linear algebraic groups" , Benjamin  (1969)</TD></TR></table>