Constructible topology

In commutative algebra, the constructible topology on the spectrum formula_1 of a commutative ring formula_2 is a topology where each closed set is the image of formula_3 in formula_1 for some algebra "B" over "A". An important feature of this construction is that the map formula_5 is a closed map with respect to the constructible topology.
With respect to this topology, formula_1 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 formula_7 is a von Neumann regular ring, where formula_8 is the nilradical of "A".
Despite the terminology being similar, the constructible topology is not the same as the set of all constructible sets.