{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $\\mathcal{N} = \\N^\\N$ be the [[Baire space|P2WS16HKD7]].\nFor each [[Polish space|584GXXY73M]] $X$, define classes of subsets of $X$ by recursion on $n \\ge 1$:\n\n1. $\\Sigma^1_1$ is the class of [[analytic subsets|3RWWZYPAFV]] of $X$.\n2. $\\Pi^1_n$ is the class of complements $X \\setminus A$ of sets $A$ in $\\Sigma^1_n$.\n3. $\\Sigma^1_{n+1}$ is the class of projections $\\set{x \\in X : \\exists y \\in \\mathcal{N} \\, (x, y) \\in C}$ of sets $C \\subseteq X \\times \\mathcal{N}$ in the class $\\Pi^1_n$ of the Polish space $X \\times \\mathcal{N}$.\n4. $\\Delta^1_n = \\Sigma^1_n \\cap \\Pi^1_n$.\n\nA subset of $X$ is *projective* if it lies in $\\Sigma^1_n$ for some $n$.\nThis system of classes is the *projective hierarchy*.\n\n## Remarks\n\nThe coanalytic sets form $\\Pi^1_1$, and by Suslin's theorem $\\Delta^1_1$ is the class of Borel sets.\nExample: the set of $x \\in \\set{0, 1}^{\\N \\times \\N}$ that code a well-order of $\\N$ is in $\\Pi^1_1$ but not in $\\Sigma^1_1$.\nNon-example by counting: there are only $2^{\\aleph_0}$ projective subsets of $\\R$, and $2^{2^{\\aleph_0}}$ subsets in all, so some subset of $\\R$ is not projective.\nEach class $\\Sigma^1_n$ and $\\Pi^1_n$ is closed under countable unions, countable intersections and preimages by continuous maps, and $\\Sigma^1_n \\cup \\Pi^1_n \\subseteq \\Delta^1_{n+1}$.\nIf $X$ is uncountable, then $\\Sigma^1_n \\neq \\Pi^1_n$ for every $n$.\nThe lightface version, defined with recursive codes instead of arbitrary parameters, is part of the [[analytical hierarchy|M6JGF4TPM6]].\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "defines": "8DZTNVTGY2",
    "kind": "definition",
    "msc": [
      "03E15",
      "54H05"
    ],
    "name": "Projective hierarchy",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Projective_hierarchy"
      }
    ],
    "tag": "NY50YEPR0X",
    "uses": [
      "584GXXY73M",
      "P2WS16HKD7",
      "3RWWZYPAFV"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/NY50YEPR0X/",
  "links": [
    {
      "field": "defines",
      "target": "8DZTNVTGY2"
    },
    {
      "field": "uses",
      "target": "584GXXY73M"
    },
    {
      "field": "uses",
      "target": "P2WS16HKD7"
    },
    {
      "field": "uses",
      "target": "3RWWZYPAFV"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/NY50YEPR0X.md",
  "msc": [
    "03E15",
    "54H05"
  ],
  "name": "Projective hierarchy",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/NY5/NY50YEPR0X.md",
  "status": null,
  "tag": "NY50YEPR0X",
  "uri": "m2://NY50YEPR0X",
  "url": "https://math2.tamnd.com/wiki/projective-hierarchy-definition/",
  "writing": {
    "tag": "NY50YEPR0X",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}