Noetherian topological space

Definition

Let $X$ be a topological space. The space $X$ is Noetherian if it satisfies the descending chain condition on closed sets. That 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 $m \ge n$ .

Remarks

Equivalent forms: every ascending chain of open subsets stabilises. Also equivalent: every nonempty family of closed subsets has a minimal element. Also equivalent: every subset of $X$ is compact in the subspace topology.

Example: any set $X$ with the cofinite topology. A proper closed subset is finite, so a strictly descending chain of closed sets has finitely many steps after its first proper member.

Non-example: $\R$ with the usual topology. The closed sets $C_n = [n, \infty)$ satisfy $C_1 \supsetneq C_2 \supsetneq \cdots$ , so the chain never stabilises.

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Tag	`M4H3QNJPGD`
URI	`m2://M4H3QNJPGD`
Kind	definition
Writing	W0 stub
MSC	54D30, 14A05, 13E05
Defines	Noetherian topological space
Uses	Topological space, Closed set
Changed	2 November 2026

Noetherian topological space

Definition

Remarks

Reading path

Further reading

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