Noetherian topological space
of closed subsets of , there is an integer such that .
As a first example, note that all finite topological spaces are Noetherian.
There is a lot of interplay between the Noetherian condition and compactness:
Example of a Noetherian topological space:
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.
|Title||Noetherian topological space|
|Date of creation||2013-03-22 13:03:33|
|Last modified on||2013-03-22 13:03:33|
|Last modified by||mathcam (2727)|