I know that a Noetherian topological space (such as Zariski topological space on Affine space) has ascending chain condition on open sets (descending chain condition on closed sets). This means that every ascending chain of open set in such a space has a maximal open set. Now, my question is: are the number of chains in such Noetherian space finite? in other words is the number of maximal open sets finite?
2026-03-26 21:27:33.1774560453
Are the number of maximal opens sets in a Noetherian topological space finite?
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No. For instance, $\Bbb A^n$ and $\Bbb P^n$ are T1, and therefore the maximal proper open sets are the complements of points.