In this post, I will prove that a Noetherian and Hausdorff topological space is finite (and therefore has the discrete topology, being Hausdorff). The proof is very short and pleasant.
Proof: Let be such a space, and suppose that it is infinite. Let be the collection of infinite closed subsets of . It is nonempty since , and therefore has a minimal member by the Noetherian assumption. Let be points of , and be disjoint open neighborhoods of and respectively (such and exist by the Hausdorff assumption). Then since and are disjoint, so . Now each of and is closed in , and is properly contained in (the first one doesn’t contain , and the second one doesn’t contain ). Therefore, by minimality of , each must be finite, and therefore is also finite, which is a contradiction.
Corollary: in any infinite Hausdorff space, there exists a strictly descending infinite chain of closed subsets . The proof above can be easily adapted to construct such a sequence.