In commutative algebra, the constructible topology on the spectrum Spec ( A ) {\displaystyle \operatorname {Spec} (A)} of a commutative ring A {\displaystyle A} is a topology where each closed set is the image of Spec ( B ) {\displaystyle \operatorname {Spec} (B)} in Spec ( A ) {\displaystyle \operatorname {Spec} (A)} for some algebra B over A. An important feature of this construction is that the map Spec ( B ) → Spec ( A ) {\displaystyle \operatorname {Spec} (B)\to \operatorname {Spec} (A)} is a closed map with respect to the constructible topology. With respect to this topology, Spec ( A ) {\displaystyle \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 / nil ( A ) {\displaystyle A/\operatorname {nil} (A)} is a von Neumann regular ring, where nil ( A ) {\displaystyle \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.
See also Constructible set (topology)
References
Atiyah, Michael Francis; Macdonald, I.G. (1969), Introduction to Commutative Algebra, Westview Press, p. 87, ISBN 978-0-201-40751-8 Knight, J. T. (1971), Commutative Algebra, Cambridge University Press, pp. 121–123, ISBN 0-521-08193-9