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.
