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  "body": "\n## Definition\n\nLet $X$ be a [[topological space|0CQHFPGJ45]].\nThe space $X$ is *Noetherian* if it satisfies the descending chain condition on [[closed sets|0ZCK8DQYN1]].\nThat is, for every sequence $C_1 \\supseteq C_2 \\supseteq C_3 \\supseteq \\cdots$ of closed subsets of $X$, there is $n$ with $C_m = C_n$ for all $m \\ge n$.\n\n## Remarks\n\nEquivalent forms: every ascending chain of open subsets stabilises.\nAlso equivalent: every nonempty family of closed subsets has a minimal element.\nAlso equivalent: every subset of $X$ is [[compact|0TVJZYB6SH]] in the subspace topology.\n\nExample: any set $X$ with the [[cofinite topology|0B906ZBTMZ]].\nA proper closed subset is finite, so a strictly descending chain of closed sets has finitely many steps after its first proper member.\n\nNon-example: $\\R$ with the usual topology.\nThe closed sets $C_n = [n, \\infty)$ satisfy $C_1 \\supsetneq C_2 \\supsetneq \\cdots$, so the chain never stabilises.\n",
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