{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "55W1M20Q3F"
    },
    {
      "field": "uses",
      "target": "6FDYJVG1CK"
    }
  ],
  "body": "\n## Definition\n\nLet $\\kappa$ be a limit [[ordinal|0RDJK7EQYY]] of uncountable [[cofinality|NTE5C260S1]], and let $C \\subseteq \\kappa$.\nThe set $C$ is a club set in $\\kappa$ if both conditions hold:\n\n1. $C$ is closed: for each nonzero limit ordinal $\\alpha \u003c \\kappa$, if $\\sup(C \\cap \\alpha) = \\alpha$, then $\\alpha \\in C$.\n2. $C$ is unbounded: for each $\\beta \u003c \\kappa$ there is $\\gamma \\in C$ with $\\gamma \\ge \\beta$.\n\nHere $\\sup$ is the [[supremum|0BTMQ04556]] of a set of ordinals.\n\n## Remarks\n\nExample: the set $C$ of limit ordinals below $\\omega_1$ is club in $\\omega_1$.\nIt is unbounded, since $\\beta + \\omega \\in C$ for each $\\beta \u003c \\omega_1$.\nIt is closed, since a limit of limit ordinals is a limit ordinal.\nNon-example: the set of successor ordinals below $\\omega_1$ is unbounded but not closed.\nIndeed the successor ordinals below $\\omega$ have supremum $\\omega$, which is not a successor.\nCondition 1 says that $C$ is closed in the order topology of $\\kappa$.\nIf $\\kappa$ is a regular uncountable cardinal, the intersection of fewer than $\\kappa$ club sets in $\\kappa$ is club.\nFor such $\\kappa$, the diagonal intersection $\\set{\\alpha \u003c \\kappa : \\alpha \\in \\bigcap_{\\xi \u003c \\alpha} C_\\xi}$ of a sequence $(C_\\xi)_{\\xi \u003c \\kappa}$ of club sets is club.\nFor such $\\kappa$, the range of a [[normal function|PXTF18X433]] from $\\kappa$ to $\\kappa$ is club, and so is its set of fixed points.\n",
  "commit": "efe85cee8dbc57d85bedbacfb97b99d1b7bee843",
  "front": {
    "defines": "YW8YSCFPG5",
    "kind": "definition",
    "msc": [
      "03E05",
      "03E10"
    ],
    "name": "Club set",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Club_set"
      }
    ],
    "tag": "GWS5QXH7WW",
    "uses": [
      "0RDJK7EQYY",
      "0BTMQ04556",
      "NTE5C260S1"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/GWS5QXH7WW/",
  "links": [
    {
      "field": "defines",
      "target": "YW8YSCFPG5"
    },
    {
      "field": "uses",
      "target": "0RDJK7EQYY"
    },
    {
      "field": "uses",
      "target": "0BTMQ04556"
    },
    {
      "field": "uses",
      "target": "NTE5C260S1"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/GWS5QXH7WW.md",
  "msc": [
    "03E05",
    "03E10"
  ],
  "name": "Club set",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/GWS/GWS5QXH7WW.md",
  "status": null,
  "tag": "GWS5QXH7WW",
  "uri": "m2://GWS5QXH7WW",
  "url": "https://math2.tamnd.com/wiki/club-set-definition/",
  "writing": {
    "tag": "GWS5QXH7WW",
    "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
      }
    ]
  }
}