{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $X$ be a [[topological space|0CQHFPGJ45]] and let $R \\subseteq X$.\nThe set $R$ is *residual* in $X$ if its complement $X \\setminus R$ is [[meagre|0XGCZ4GDZ6]] in $X$.\n\n## Remarks\n\nEquivalently, $R$ contains a set $\\bigcap_{n \\ge 1} U_n$, where each $U_n$ is open and [[dense|0BSJSX7ME2]] in $X$.\nResidual sets are also called comeagre sets.\n\nExample: the set of irrational numbers is residual in $\\R$.\nIts complement $\\Q$ is countable, so it is a countable union of single points, and each single point is [[nowhere dense|0VQYCCQVJ5]].\n\nNon-example: $\\Q$ is not residual in $\\R$.\nOtherwise $\\R$ would be the union of the meagre sets $\\Q$ and $\\R \\setminus \\Q$.\nThis contradicts the [[Baire category theorem|0A7KGAYHGV]], since $\\R$ is a complete metric space.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "defines": "GRH8MCVN3N",
    "kind": "definition",
    "msc": [
      "54E52"
    ],
    "name": "Residual set",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Meagre_set"
      }
    ],
    "tag": "PDDAJEYDE9",
    "uses": [
      "0CQHFPGJ45",
      "0XGCZ4GDZ6",
      "0BSJSX7ME2"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/PDDAJEYDE9/",
  "links": [
    {
      "field": "defines",
      "target": "GRH8MCVN3N"
    },
    {
      "field": "uses",
      "target": "0CQHFPGJ45"
    },
    {
      "field": "uses",
      "target": "0XGCZ4GDZ6"
    },
    {
      "field": "uses",
      "target": "0BSJSX7ME2"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/PDDAJEYDE9.md",
  "msc": [
    "54E52"
  ],
  "name": "Residual set",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/PDD/PDDAJEYDE9.md",
  "status": null,
  "tag": "PDDAJEYDE9",
  "uri": "m2://PDDAJEYDE9",
  "url": "https://math2.tamnd.com/wiki/residual-set-definition/",
  "writing": {
    "tag": "PDDAJEYDE9",
    "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
      }
    ]
  }
}