Constructible set (topology)

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 ${\displaystyle f:X\to Y}$ 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 map ${\displaystyle \mathbf {A} ^{2}\rightarrow \mathbf {A} ^{2}}$ that sends ${\displaystyle (x,y)}$ to ${\displaystyle (x,xy)}$ has image the set ${\displaystyle \{x\neq 0\}\cup \{x=y=0\}}$, which is not a variety, but is constructible.

In any (not necessarily Noetherian) topological space, every constructible set contains a dense open subset of its closure.^[1]

Warning: In EGA III, Def.9.1.2, constructible sets are defined using only retrocompact opens. That is, the family of constructible sets of a topological space is defined as the smallest family closed under finite intersection and complement and containing all retrocompact open subsets.

So for example, the origin ${\displaystyle V(x_{1},x_{2},x_{3},\dots )}$ in the infinite affine space ${\displaystyle \mathbb {A} ^{\infty }=Spec(k[x_{1},x_{2},x_{3},\dots ])}$ is not constructible.

In any locally noetherian topological space, all subsets are retrocompact (EGA III, 9.1), so the two definitions are the same in this setting.

See also[]

• Constructible topology
• Constructible sheaf

Notes[]

References[]

• Allouche, Jean Paul. Note on the constructible sets of a topological space.
• Andradas, Carlos; Bröcker, Ludwig; Ruiz, Jesús M. (1996). Constructible sets in real geometry. Ergebnisse der Mathematik und ihrer Grenzgebiete (3) --- Results in Mathematics and Related Areas (3). 33. Berlin: Springer-Verlag. pp. x+270. ISBN 3-540-60451-0. MR 1393194.
• Borel, Armand. Linear algebraic groups.
• Grothendieck, Alexander. EGA 0 §9
• Grothendieck, Alexandre; Dieudonné, Jean (1971). Éléments de géométrie algébrique: I. Le langage des schémas. Grundlehren der Mathematischen Wissenschaften (in French). 166 (2nd ed.). Berlin; New York: Springer-Verlag. ISBN 978-3-540-05113-8.
• Grothendieck, Alexandre; Dieudonné, Jean (1960). "Éléments de géométrie algébrique: I. Le langage des schémas". Publications Mathématiques de l'IHÉS. 4: 5–228. doi:10.1007/bf02684778. MR 0217083.
• Mostowski, A. (1969). Constructible sets with applications. Studies in Logic and the Foundations of Mathematics. Amsterdam --- Warsaw: North-Holland Publishing Co. ---- PWN-Polish Scientific Publishers. pp. ix+269. MR 0255390.