nLab constructible set

Idea

Constructible sets are one of the central notions in descriptive set theory.

Definition

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.

Examples

• In general topology, given a topological space $(X,\tau)$, a constructible subset of $X$, sometimes (when $(X,\tau)$ is fixed) often simply referred as a constructible set, is an element of the smallest Boolean algebra of sets containing the topology $\tau$; 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 $K$, the constructible sets in the affine space $K^n$ are precisely the definable sets for the language of fields.

Related concepts

• constructible universe

• recursive subset, definable set

• constructible sheaf

References

• Wikipedia, Constructible set