{
  "aliases": null,
  "backlinks": [
    {
      "field": "defines",
      "target": "GWS5QXH7WW"
    }
  ],
  "body": "\n## Summary\n\nA club set is a subset of a limit ordinal that is closed under limits and unbounded, and the name abbreviates \"closed unbounded\".\nBelow a [[regular cardinal|13ZV2F8JAR]] $\\kappa \u003e \\omega$, the club sets play the role of \"large\" sets: the intersection of fewer than $\\kappa$ of them is again club.\nThey define the [[stationary sets|YE6WZR46RQ]], and the [[precise definition|GWS5QXH7WW]] is given separately.\n\n## Explanations\n\n### L1 human\n\nThink of the countable ordinals as a very long line.\nA club set is a subset that keeps appearing as far as you go, and that contains every point it approaches from below.\nFor example, the limit ordinals below $\\omega_1$, such as $\\omega$, $\\omega \\cdot 2$ and $\\omega^2$, form a club set.\nThe successor ordinals do not: they keep appearing, but the limits they approach, like $\\omega$, are not successors.\nTwo club sets of countable ordinals always meet, and in fact their intersection is again club.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "kind": "concept",
    "msc": [
      "03E05",
      "03E10"
    ],
    "name": "Club set",
    "same_as": {
      "wikidata": "Q310647",
      "wikipedia": "Club set"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Club_set"
      }
    ],
    "tag": "YW8YSCFPG5"
  },
  "kind": "concept",
  "link": "https://math2.tamnd.com/node/YW8YSCFPG5/",
  "links": null,
  "markdown": "https://math2.tamnd.com/node/YW8YSCFPG5.md",
  "msc": [
    "03E05",
    "03E10"
  ],
  "name": "Club set",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/YW8/YW8YSCFPG5.md",
  "status": null,
  "tag": "YW8YSCFPG5",
  "uri": "m2://YW8YSCFPG5",
  "url": "https://math2.tamnd.com/wiki/club-set/",
  "writing": {
    "tag": "YW8YSCFPG5",
    "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
      }
    ]
  }
}