{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $(X, \\tau, \\le)$ be a [[Priestley space|ZGBV8V3823]].\nSo $(X, \\tau)$ is a compact topological space and $\\le$ is a [[partial order|0QV2JS9RJ3]] on $X$.\nAlso, for all $x, y \\in X$ with $x \\not\\le y$, there is a [[clopen|DQ4E55T3BW]] [[up-set|ETCE287E2R]] $U$ with $x \\in U$ and $y \\notin U$.\nFor $S \\subseteq X$, write ${\\downarrow} S = \\set{x \\in X : x \\le s \\text{ for some } s \\in S}$.\nThe space $(X, \\tau, \\le)$ is an *Esakia space* if ${\\downarrow} C$ is clopen for every clopen set $C \\subseteq X$.\n\n## Remarks\n\nIn a Priestley space, ${\\downarrow} C$ is always closed when $C$ is closed; the extra condition is that it is also open when $C$ is clopen.\nFor clopen up-sets $U$ and $V$ of an Esakia space, the set $X \\setminus {\\downarrow}(U \\setminus V)$ is a clopen up-set, and it is the Heyting implication $U \\to V$.\n\nExample: a finite poset $(P, \\le)$ with the discrete topology.\nEvery subset is clopen, so the separation condition holds with $U = {\\uparrow} x$, and every down-set is clopen.\n\nNon-example: let $X = \\set{a_1, a_2, \\ldots} \\cup \\set{\\infty, b}$, where $a_n \\to \\infty$ as in the one-point compactification of $\\N$, and $b$ is an isolated point.\nOrder $X$ by $\\infty \\le b$ and no other strict relations.\nThis is a Priestley space, but $\\set{b}$ is clopen and ${\\downarrow}\\set{b} = \\set{\\infty, b}$ is not open, since every neighbourhood of $\\infty$ contains some $a_n$.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "996BWM6ZQ7",
    "kind": "definition",
    "msc": [
      "06D20",
      "06D50",
      "03G10"
    ],
    "name": "Esakia space",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Esakia_space"
      }
    ],
    "tag": "59PG49KH6V",
    "uses": [
      "ZGBV8V3823",
      "DQ4E55T3BW",
      "0QV2JS9RJ3",
      "ETCE287E2R"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/59PG49KH6V/",
  "links": [
    {
      "field": "defines",
      "target": "996BWM6ZQ7"
    },
    {
      "field": "uses",
      "target": "ZGBV8V3823"
    },
    {
      "field": "uses",
      "target": "DQ4E55T3BW"
    },
    {
      "field": "uses",
      "target": "0QV2JS9RJ3"
    },
    {
      "field": "uses",
      "target": "ETCE287E2R"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/59PG49KH6V.md",
  "msc": [
    "06D20",
    "06D50",
    "03G10"
  ],
  "name": "Esakia space",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/59P/59PG49KH6V.md",
  "status": null,
  "tag": "59PG49KH6V",
  "uri": "m2://59PG49KH6V",
  "url": "https://tamnd.github.io/math2/wiki/esakia-space-definition/",
  "writing": {
    "tag": "59PG49KH6V",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      },
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}