Example
The space (affine -space over a field ) under the Zariski topology is an example of a Noetherian topological space. By properties of the ideal of a subset of, we know that if
is a descending chain of Zariski-closed subsets, then
is an ascending chain of ideals of Since is a Noetherian ring, there exists an integer such that
But because we have a one-to-one correspondence between radical ideals of and Zariski-closed sets in we have for all Hence
- as required.
Read more about this topic: Noetherian Topological Space
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“Our intellect is not the most subtle, the most powerful, the most appropriate, instrument for revealing the truth. It is life that, little by little, example by example, permits us to see that what is most important to our heart, or to our mind, is learned not by reasoning but through other agencies. Then it is that the intellect, observing their superiority, abdicates its control to them upon reasoned grounds and agrees to become their collaborator and lackey.”
—Marcel Proust (1871–1922)