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 \( {\displaystyle \operatorname {Spec} (B)} \) in \( \operatorname {Spec}(A) \) for some algebra B over A. An important feature of this construction is that the map \( {\displaystyle \operatorname {Spec} (B)\to \operatorname {Spec} (A)} \) is a closed map with respect to the constructible topology.
With respect to this topology, \( \operatorname {Spec}(A) \) is a compact,[1] 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 \( {\displaystyle A/\operatorname {nil} (A)} \) is a von Neumann regular ring, where \( {\displaystyle \operatorname {nil} (A)} \) is the nilradical of A.
Despite the terminology being similar, the constructible topology is not the same as the set of all constructible sets.[2]
See also
Constructible set (topology)
References
Some authors prefer the term quasicompact here.
"Reconciling two different definitions of constructible sets". math.stackexchange.com. Retrieved 2016-10-13.
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
Undergraduate Texts in Mathematics
Graduate Studies in Mathematics
Hellenica World - Scientific Library
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