Noetherian topological space

A topological spaceMathworldPlanetmath X is called if it satisfies the descending chain conditionMathworldPlanetmathPlanetmath for closed subsets: for any sequencePlanetmathPlanetmath


of closed subsets Yi of X, there is an integer m such that Ym=Ym+1=.

As a first example, note that all finite topological spaces are NoetherianPlanetmathPlanetmathPlanetmath.

There is a lot of interplay between the Noetherian condition and compactness:

Note that if R is a Noetherian ring, then Spec(R), the prime spectrum of R, is a Noetherian topological space.

Example of a Noetherian topological space:
The space 𝔸kn (affine n-space over a field k) under the Zariski topologyMathworldPlanetmath is an example of a Noetherian topological space. By properties of the ideal of a subset of 𝔸kn, we know that if Y1Y2 is a descending chain of Zariski-closed subsets, then I(Y1)I(Y2) is an ascending chain of ideals of k[x1,,xn].

Since k[x1,,xn] is a Noetherian ring, there exists an integer m such that I(Ym)=I(Ym+1)=. But because we have a one-to-one correspondence between radical ideals of k[x1,,xn] and Zariski-closed sets in 𝔸kn, we have V(I(Yi))=Yi for all i. Hence Ym=Ym+1= as required.

Title Noetherian topological space
Canonical name NoetherianTopologicalSpace
Date of creation 2013-03-22 13:03:33
Last modified on 2013-03-22 13:03:33
Owner mathcam (2727)
Last modified by mathcam (2727)
Numerical id 18
Author mathcam (2727)
Entry type Definition
Classification msc 14A10
Related topic Compact