Constructible sets are one of the central notions in descriptive set theory?.
In set theory, a (Gödel) constructible set is any set in a constructible universe, see there.
One can generalize to constructible elements in more general Boolean lattices.
In general topology, given a topological space , a constructible subset of , sometimes (when is fixed) often simply referred as a constructible set, is an element of the smallest Boolean algebra of sets containing the topology ; the constructible sets are precisely the finite unions of locally closed sets.
In the case of theory ACF of algebraically closed fields?, for every model , the constructible sets in the affine space are precisely the definable sets for the language? of fields.