{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "AGYBSTD7BG"
    },
    {
      "field": "uses",
      "target": "DT5PH9QVQ7"
    },
    {
      "field": "uses",
      "target": "HBBE0C1NFD"
    },
    {
      "field": "uses",
      "target": "TK6R2C9KRE"
    },
    {
      "field": "uses",
      "target": "VFPGADHJYA"
    },
    {
      "field": "uses",
      "target": "X117TDBR51"
    },
    {
      "field": "uses",
      "target": "YYP6QNRERY"
    },
    {
      "field": "about",
      "target": "DT5PH9QVQ7"
    },
    {
      "field": "defines",
      "target": "SR56KCFYYS"
    }
  ],
  "body": "\n## Summary\n\nAn ultrafilter on a set $X$ is a [[filter|15PF73BKY6]] on $X$ that is maximal: no strictly larger filter on $X$ exists.\nEquivalently, for every subset $A$ of $X$, either $A$ or its complement belongs to it.\nEvery filter lies in an ultrafilter, by [[Zorn's lemma|04RXTQA5ZA]].\nUltrafilters describe [[compactness|0TVJZYB6SH]], build the Stone–Čech compactification, and give ultraproducts in logic.\n\n## Explanations\n\n### L1 human\n\nAn ultrafilter makes a firm decision about every subset: either the set is \"big\" or its complement is.\nThe simplest ultrafilters pick a point $a$ and call a set big when it contains $a$.\nOn an infinite set there are also ultrafilters that are not of this kind.\nOne can show that they exist, using the axiom of choice, but one cannot write one down explicitly.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "kind": "concept",
    "msc": [
      "54A20",
      "03E05"
    ],
    "name": "Ultrafilter",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Ultrafilter"
      }
    ],
    "specializes": [
      "15PF73BKY6"
    ],
    "tag": "J64Y3JC0RE"
  },
  "kind": "concept",
  "link": "https://math2.tamnd.com/node/J64Y3JC0RE/",
  "links": [
    {
      "field": "specializes",
      "target": "15PF73BKY6"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/J64Y3JC0RE.md",
  "msc": [
    "54A20",
    "03E05"
  ],
  "name": "Ultrafilter",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/J64/J64Y3JC0RE.md",
  "status": null,
  "tag": "J64Y3JC0RE",
  "uri": "m2://J64Y3JC0RE",
  "url": "https://math2.tamnd.com/wiki/ultrafilter/",
  "writing": {
    "tag": "J64Y3JC0RE",
    "kind": "concept",
    "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
      }
    ]
  }
}