{
  "aliases": null,
  "backlinks": [
    {
      "field": "defines",
      "target": "8APGCB1CKH"
    },
    {
      "field": "uses",
      "target": "9D26WKTFPT"
    }
  ],
  "body": "\n## Summary\n\nThe specialization preorder of a topological space orders its points by closure: $x \\le y$ when $x$ lies in the closure of $\\set{y}$.\nIt is a partial order exactly when the space is [[T0|07SG12TQH3]], and it is the equality relation exactly when the space is T1.\nIn algebraic geometry it records which points of a scheme specialize to which.\nIn domain theory it recovers the order of a poset from its Scott topology, and in general it links topological spaces with ordered sets.\n\n## Explanations\n\n### L1 human\n\nIn a nice space like the plane, every point is closed and the specialization preorder says nothing.\nIn spaces that are not T1 some points sit \"inside the closure\" of others, and the preorder records this.\nIn the Sierpiński space with points $0$ and $1$ and open sets $\\emptyset$, $\\set{1}$ and $\\set{0, 1}$, the closure of $\\set{1}$ is $\\set{0, 1}$.\nSo $0 \\le 1$: the point $1$ is a \"general\" point and $0$ is a \"special\" one.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "54D10",
      "06A06"
    ],
    "name": "Specialization preorder",
    "same_as": {
      "wikidata": "Q5569420",
      "wikipedia": "Specialization (pre)order"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Specialization_(pre)order"
      }
    ],
    "specializes": [
      "0VABYWJWH9"
    ],
    "tag": "CMYVK9CEKZ"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/CMYVK9CEKZ/",
  "links": [
    {
      "field": "specializes",
      "target": "0VABYWJWH9"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/CMYVK9CEKZ.md",
  "msc": [
    "54D10",
    "06A06"
  ],
  "name": "Specialization preorder",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/CMY/CMYVK9CEKZ.md",
  "status": null,
  "tag": "CMYVK9CEKZ",
  "uri": "m2://CMYVK9CEKZ",
  "url": "https://tamnd.github.io/math2/wiki/specialization-preorder/",
  "writing": {
    "tag": "CMYVK9CEKZ",
    "kind": "concept",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}