Constructible set (topology)


In topology, a constructible set in a topological space is a finite union of locally closed sets. Constructible sets form a Boolean algebra 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 states: Let 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 always a constructible set. For example, the map that sends to has image the set, which is not a variety, but is constructible.
In any topological space, every constructible set contains a dense open subset of its closure.
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 in the infinite affine space is not constructible.
In any locally noetherian topological space, all subsets are retrocompact, so the two definitions are the same in this setting.