In commutative algebra, the constructible topology on the spectrum Spec ( A ) of a commutative ring A is a topology where each closed set is the image of Spec ( B ) in Spec ( A ) for some algebra B over A. An important feature of this construction is that the map Spec ( B ) → Spec ( A ) is a closed map with respect to the constructible topology.
With respect to this topology, Spec ( A ) is a compact, Hausdorff, and totally disconnected topological space. In general the constructible topology is a finer topology than the Zariski topology, but the two topologies will coincide if and only if A / nil ( A ) is a von Neumann regular ring, where 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.
