---
tag: 2VQX2E8EAH
kind: concept
name: Noetherian topological space
msc: [54D30, 14A05, "13E05"]
specializes: [0TVJZYB6SH]
sources:
  - url: https://en.wikipedia.org/wiki/Noetherian_topological_space
same_as:
  wikidata: Q590554
  wikipedia: Noetherian topological space
---

## Summary

A Noetherian topological space is a [[topological space|0CQHFPGJ45]] in which every descending chain of closed subsets eventually stops.
It is the topological counterpart of a [[Noetherian ring|0670T59HGE]]: the spectrum of a Noetherian ring with the Zariski topology is a Noetherian space.
Every Noetherian space is [[compact|0TVJZYB6SH]], and so is each of its subspaces.
Every Noetherian space is a finite union of [[irreducible|38CDCK1RYB]] closed subsets, its irreducible components.

## Explanations

### L1 human

In a Noetherian space you cannot keep cutting closed sets smaller and smaller forever.
Any chain of shrinking closed sets must become constant after finitely many steps.
Finite spaces are Noetherian, and so is any set with the [[cofinite topology|0B906ZBTMZ]].
The real line is not: the closed sets $[n, \infty)$ for $n = 1, 2, 3, \dots$ shrink forever.