{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $X$ be a [[Polish space|584GXXY73M]], and let $\\omega_1$ be the first uncountable [[ordinal|0YSCSX5YZC]].\nDefine classes of subsets of $X$ by recursion on $\\alpha$ with $1 \\le \\alpha \u003c \\omega_1$:\n\n1. $\\Sigma^0_1$ is the class of open subsets of $X$.\n2. For $\\alpha \\ge 1$, $\\Pi^0_\\alpha$ is the class of complements $X \\setminus A$ of sets $A$ in $\\Sigma^0_\\alpha$.\n3. For $\\alpha \u003e 1$, $\\Sigma^0_\\alpha$ is the class of countable unions $\\bigcup_{n} A_n$ with each $A_n$ in $\\Pi^0_{\\beta_n}$ for some $\\beta_n \u003c \\alpha$.\n4. For $\\alpha \\ge 1$, $\\Delta^0_\\alpha = \\Sigma^0_\\alpha \\cap \\Pi^0_\\alpha$.\n\nThis system of classes is the *Borel hierarchy* of $X$.\n\n## Remarks\n\nThe class $\\Pi^0_1$ is the class of closed sets, $\\Sigma^0_2$ is the class of $F_\\sigma$ sets, and $\\Pi^0_2$ is the class of $G_\\delta$ sets.\nIn $X = \\R$, the set $\\Q$ is in $\\Sigma^0_2$, as the countable union of the closed sets $\\set{q}$.\nIt is not in $\\Pi^0_2$, since by the Baire category theorem a dense $G_\\delta$ subset of $\\R$ is uncountable.\nEvery class satisfies $\\Sigma^0_\\alpha \\cup \\Pi^0_\\alpha \\subseteq \\Delta^0_{\\alpha + 1}$, and the union of all the classes $\\Sigma^0_\\alpha$ is the [[Borel sigma-algebra|0WQXY32Q8D]] of $X$.\nA non-example: an analytic set that is not Borel lies in no class of the hierarchy.\nIf $X$ is uncountable, then $\\Sigma^0_\\alpha \\neq \\Pi^0_\\alpha$ for every $\\alpha$, so the hierarchy does not collapse.\nThe same definition works in any metrizable space; metrizability is used to show that closed sets are $G_\\delta$ sets.\n",
  "commit": "efe85cee8dbc57d85bedbacfb97b99d1b7bee843",
  "front": {
    "defines": "4P4Z4ZGXCY",
    "kind": "definition",
    "msc": [
      "03E15",
      "54H05",
      "28A05"
    ],
    "name": "Borel hierarchy",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Borel_hierarchy"
      }
    ],
    "tag": "RMH7ZQ2F40",
    "uses": [
      "584GXXY73M",
      "0YSCSX5YZC",
      "0WQXY32Q8D"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/RMH7ZQ2F40/",
  "links": [
    {
      "field": "defines",
      "target": "4P4Z4ZGXCY"
    },
    {
      "field": "uses",
      "target": "584GXXY73M"
    },
    {
      "field": "uses",
      "target": "0YSCSX5YZC"
    },
    {
      "field": "uses",
      "target": "0WQXY32Q8D"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/RMH7ZQ2F40.md",
  "msc": [
    "03E15",
    "54H05",
    "28A05"
  ],
  "name": "Borel hierarchy",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/RMH/RMH7ZQ2F40.md",
  "status": null,
  "tag": "RMH7ZQ2F40",
  "uri": "m2://RMH7ZQ2F40",
  "url": "https://math2.tamnd.com/wiki/borel-hierarchy-definition/",
  "writing": {
    "tag": "RMH7ZQ2F40",
    "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
      }
    ]
  }
}