{
  "aliases": null,
  "backlinks": [
    {
      "field": "defines",
      "target": "M4H3QNJPGD"
    }
  ],
  "body": "\n## Summary\n\nA Noetherian topological space is a [[topological space|0CQHFPGJ45]] in which every descending chain of closed subsets eventually stops.\nIt is the topological counterpart of a [[Noetherian ring|0670T59HGE]]: the spectrum of a Noetherian ring with the Zariski topology is a Noetherian space.\nEvery Noetherian space is [[compact|0TVJZYB6SH]], and so is each of its subspaces.\nEvery Noetherian space is a finite union of [[irreducible|38CDCK1RYB]] closed subsets, its irreducible components.\n\n## Explanations\n\n### L1 human\n\nIn a Noetherian space you cannot keep cutting closed sets smaller and smaller forever.\nAny chain of shrinking closed sets must become constant after finitely many steps.\nFinite spaces are Noetherian, and so is any set with the [[cofinite topology|0B906ZBTMZ]].\nThe real line is not: the closed sets $[n, \\infty)$ for $n = 1, 2, 3, \\dots$ shrink forever.\n",
  "commit": "5fc425f97f1f6b11da5892c2948ef8c5d8805665",
  "front": {
    "kind": "concept",
    "msc": [
      "54D30",
      "14A05",
      "13E05"
    ],
    "name": "Noetherian topological space",
    "same_as": {
      "wikidata": "Q590554",
      "wikipedia": "Noetherian topological space"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Noetherian_topological_space"
      }
    ],
    "specializes": [
      "0TVJZYB6SH"
    ],
    "tag": "2VQX2E8EAH"
  },
  "kind": "concept",
  "link": "https://math2.tamnd.com/node/2VQX2E8EAH/",
  "links": [
    {
      "field": "specializes",
      "target": "0TVJZYB6SH"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/2VQX2E8EAH.md",
  "msc": [
    "54D30",
    "14A05",
    "13E05"
  ],
  "name": "Noetherian topological space",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/2VQ/2VQX2E8EAH.md",
  "status": null,
  "tag": "2VQX2E8EAH",
  "uri": "m2://2VQX2E8EAH",
  "url": "https://math2.tamnd.com/wiki/noetherian-topological-space/",
  "writing": {
    "tag": "2VQX2E8EAH",
    "kind": "concept",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}