{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $X$ be a [[topological space|0CQHFPGJ45]].\nA subset $C \\subseteq X$ is irreducible when $C \\ne \\varnothing$ and the following holds.\nIf $C = A \\cup B$ with $A$ and $B$ [[closed|0ZCK8DQYN1]] in the subspace $C$, then $A = C$ or $B = C$.\nA point $x$ is a generic point of a closed set $C$ when the [[closure|0QEAJB91GZ]] $\\overline{\\set{x}}$ equals $C$.\n$X$ is *sober* when every irreducible closed subset of $X$ has exactly one generic point.\n\n## Remarks\n\nFor each point $x$, the closure $\\overline{\\set{x}}$ is irreducible and closed, with generic point $x$.\nSo the uniqueness part says that $\\overline{\\set{x}} = \\overline{\\set{y}}$ implies $x = y$, and a sober space is [[T0|07SG12TQH3]].\nEvery [[Hausdorff space|0S1FJ8VEX9]] is sober, because its irreducible subsets are single points.\n\nExample: for a commutative ring $R$, the spectrum $\\operatorname{Spec} R$ with the Zariski topology is sober.\nAn irreducible closed set has the form $V(\\mathfrak p) = \\set{\\mathfrak q : \\mathfrak q \\supseteq \\mathfrak p}$ for a unique [[prime ideal|00MHHZ9SNE]] $\\mathfrak p$, and $\\mathfrak p$ is its generic point.\n\nNon-example: an infinite set $X$ with the [[cofinite topology|0B906ZBTMZ]] is not sober.\n$X$ is irreducible, since two proper closed sets are finite and their union is not $X$.\nBut each point is closed, so $\\overline{\\set{x}} = \\set{x} \\ne X$, and $X$ has no generic point.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "defines": "4N4DNDR22Z",
    "kind": "definition",
    "msc": [
      "54D10",
      "06D22"
    ],
    "name": "Sober space",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Sober_space"
      }
    ],
    "tag": "N8ZY3A50HK",
    "uses": [
      "0CQHFPGJ45",
      "0ZCK8DQYN1",
      "0QEAJB91GZ",
      "07SG12TQH3",
      "0S1FJ8VEX9",
      "00MHHZ9SNE"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/N8ZY3A50HK/",
  "links": [
    {
      "field": "defines",
      "target": "4N4DNDR22Z"
    },
    {
      "field": "uses",
      "target": "0CQHFPGJ45"
    },
    {
      "field": "uses",
      "target": "0ZCK8DQYN1"
    },
    {
      "field": "uses",
      "target": "0QEAJB91GZ"
    },
    {
      "field": "uses",
      "target": "07SG12TQH3"
    },
    {
      "field": "uses",
      "target": "0S1FJ8VEX9"
    },
    {
      "field": "uses",
      "target": "00MHHZ9SNE"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/N8ZY3A50HK.md",
  "msc": [
    "54D10",
    "06D22"
  ],
  "name": "Sober space",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/N8Z/N8ZY3A50HK.md",
  "status": null,
  "tag": "N8ZY3A50HK",
  "uri": "m2://N8ZY3A50HK",
  "url": "https://math2.tamnd.com/wiki/sober-space-definition/",
  "writing": {
    "tag": "N8ZY3A50HK",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}