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# A Noetherian and Hausdorff space is finite

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 ${X}$ be such a space, and suppose that it is infinite. Let ${\Sigma}$ be the collection of infinite closed subsets of ${X}$. It is nonempty since ${X \in \Sigma}$, and therefore has a minimal member ${Z}$ by the Noetherian assumption. Let ${p,q}$ be points of ${Z}$, and ${U,V}$ be disjoint open neighborhoods of ${p}$ and ${q}$ respectively (such ${U}$ and ${V}$ exist by the Hausdorff assumption). Then ${X = (X-U) \cup (X-V)}$ since ${U}$ and ${V}$ are disjoint, so ${Z = (Z \cap (X-U)) \cup (Z \cap (X-V))}$. Now each of ${Z \cap (X-U)}$ and ${Z \cap (X-V)}$ is closed in ${X}$, and is properly contained in ${Z}$ (the first one doesn’t contain ${p}$, and the second one doesn’t contain ${q}$). Therefore, by minimality of ${Z}$, each must be finite, and therefore ${Z}$ is also finite, which is a contradiction. $\Box$

Corollary: in any infinite Hausdorff space, there exists a strictly descending infinite chain of closed subsets $Z_1 \supset Z_2 \supset Z_3 \dots$. The proof above can be easily adapted to construct such a sequence.