Constructible topology

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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 \operatorname{Spec} (B) in \operatorname{Spec}(A) for some algebra B over A. An important feature of this construction is that the map \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 A / \operatorname{nil}(A) is a von Neumann regular ring, where \operatorname{nil}(A)\, is the nilradical of A.

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  1. ^ Some authors prefer the term quasicompact here.