Constructible topology

In commutative algebra, the constructible topology on the spectrum $$\operatorname{Spec}(A)$$ of a commutative ring $$A$$ is a topology where each closed set is the image of $$\operatorname{Spec} (B)$$ in $$\operatorname{Spec}(A)$$ for some algebra B over A. An important feature of this construction is that the map $$\operatorname{Spec}(B) \to \operatorname{Spec}(A)$$ is a closed map with respect to the constructible topology.

With respect to this topology, $$\operatorname{Spec}(A)$$ is a compact, Hausdorff, and totally disconnected topological space (i.e., a Stone space). In general, the constructible topology is a finer topology than the Zariski topology, and the two topologies coincide if and only if $$A / \operatorname{nil}(A)$$ is a von Neumann regular ring, where $$\operatorname{nil}(A)$$ is the nilradical of A.

Despite the terminology being similar, the constructible topology is not the same as the set of all constructible sets.