{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $X$ be a [[topological space|0CNQXVBHCX]] and let $A \\subseteq X$.\nThe set $A$ is meagre in $X$ when there is a [[countable|0ZG61TFBES]] family $(A_n)_{n \\in \\mathbb{N}}$ of [[nowhere dense|0KSM23SN3H]] subsets of $X$ with $A = \\bigcup_{n} A_n$.\n\n## Remarks\n\nAnother name for a meagre set is a set of first category. A set that is not meagre is of second category.\nEach subset of a meagre set is meagre, and each countable union of meagre sets is meagre.\nThe set $\\mathbb{Q}$ is meagre in $\\mathbb{R}$, because it is a countable union of one-point sets.\nBy the [[Baire category theorem|0A7KGAYHGV]], $\\mathbb{R}$ itself is not meagre in $\\mathbb{R}$.\n\n## Explanations\n\n### L1 human\n\nA meagre set is made of countably many thin pieces.\nIt can still be dense, as $\\mathbb{Q}$ is, but in a complete metric space it can never fill an open set.\n",
  "commit": "6ac761d40018146541484a9e8b9eafd6ecc303dc",
  "front": {
    "defines": "0XGCZ4GDZ6",
    "kind": "definition",
    "msc": [
      "54E52"
    ],
    "name": "Meagre set",
    "sources": [
      {
        "locator": "Definitions",
        "url": "https://en.wikipedia.org/wiki/Meagre_set"
      }
    ],
    "tag": "0FQPYAY9PJ",
    "uses": [
      "0KSM23SN3H",
      "0ZG61TFBES"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/0FQPYAY9PJ/",
  "links": [
    {
      "field": "defines",
      "target": "0XGCZ4GDZ6"
    },
    {
      "field": "uses",
      "target": "0KSM23SN3H"
    },
    {
      "field": "uses",
      "target": "0ZG61TFBES"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/0FQPYAY9PJ.md",
  "msc": [
    "54E52"
  ],
  "name": "Meagre set",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/0FQ/0FQPYAY9PJ.md",
  "status": null,
  "tag": "0FQPYAY9PJ",
  "uri": "m2://0FQPYAY9PJ",
  "url": "https://math2.tamnd.com/wiki/meagre-set-definition/",
  "writing": {
    "tag": "0FQPYAY9PJ",
    "kind": "definition",
    "level": "W1",
    "next": [
      {
        "what": "edge instance_of in",
        "note": "one example",
        "have": 0,
        "want": 1
      },
      {
        "what": "edge not_instance_of in",
        "note": "one non-example that fails one clause",
        "have": 0,
        "want": 1
      }
    ]
  }
}