Constructible set (topology)

In topology, constructible sets are a class of subsets of a topological space that have a relatively "simple" structure. They are used particularly in algebraic geometry and related fields. A key result known as Chevalley's theorem in algebraic geometry shows that the image of a constructible set is constructible for an important class of mappings (more specifically morphisms) of algebraic varieties (or more generally schemes). In addition, a large number of "local" geometric properties of schemes, morphisms and sheaves are (locally) constructible. Constructible sets also feature in the definition of various types of constructible sheaves in algebraic geometry and intersection cohomology. A simple definition, adequate in many situations, is that a constructible set 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.) However, a modification and another slightly weaker definition are needed to have definitions that behave better with "large" spaces: Definitions: A subset of a topological space is called retrocompact if is compact for every compact open subset . A subset of is constructible if it is a finite union of subsets of the form where both and are open and retrocompact subsets of . A subset is locally constructible if there is a cover of consisting of open subsets with the property that each is a constructible subset of . Equivalently the constructible subsets of a topological space are the smallest collection of subsets of that (i) contains all open retrocompact subsets and (ii) contains all complements and finite unions (and hence also finite intersections) of sets in it. In other words, constructible sets are precisely the Boolean algebra generated by retrocompact open subsets. In a locally noetherian topological space, all subsets are retrocompact, and so for such spaces the simplified definition given first above is equivalent to the more elaborate one.