Constructible set (topology)

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 formula_1 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 formula_2 that sends formula_3 to formula_4 has image the set formula_5, 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.
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 formula_6 in the infinite affine space formula_7 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.