{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $X$ be a [[topological space|0CQHFPGJ45]] and let $A \\subseteq X$.\nThe *derived set* of $A$ is the set $A'$ of all [[limit points|0E3X7TXS67]] of $A$.\nSo $x \\in A'$ if and only if every [[neighbourhood|0ZCY0Z80B6]] of $x$ contains a point of $A$ different from $x$.\n\n## Remarks\n\nThe [[closure|0QEAJB91GZ]] of $A$ is $\\overline{A} = A \\cup A'$.\nIn a [[T1 space|0NW6CH38YR]], the derived set $A'$ is always [[closed|0ZCK8DQYN1]].\n\nExample: in $\\R$, let $A = \\set{1/n : n \\ge 1}$.\nEach $1/n$ has a neighbourhood that meets $A$ only in $1/n$, so $1/n \\notin A'$.\nEvery interval $(-\\varepsilon, \\varepsilon)$ contains $1/n$ for large $n$, so $0 \\in A'$.\nThus $A' = \\set{0}$.\n\nExample: the derived set of $\\Q$ in $\\R$ is $\\R$, and the derived set of $\\Z$ in $\\R$ is $\\emptyset$.\n\nNon-example: in $\\R$, the point $2$ is not in the derived set of $A = \\set{2} \\cup [3, 4]$.\nThe neighbourhood $(1, 5/2)$ of $2$ contains no point of $A$ other than $2$, so the condition fails.\n",
  "commit": "5fc425f97f1f6b11da5892c2948ef8c5d8805665",
  "front": {
    "defines": "8ZAPAPYJFG",
    "kind": "definition",
    "msc": [
      "54A05"
    ],
    "name": "Derived set",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Derived_set_(mathematics)"
      }
    ],
    "tag": "JPQM16D84G",
    "uses": [
      "0CQHFPGJ45",
      "0E3X7TXS67",
      "0ZCY0Z80B6",
      "0QEAJB91GZ"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/JPQM16D84G/",
  "links": [
    {
      "field": "defines",
      "target": "8ZAPAPYJFG"
    },
    {
      "field": "uses",
      "target": "0CQHFPGJ45"
    },
    {
      "field": "uses",
      "target": "0E3X7TXS67"
    },
    {
      "field": "uses",
      "target": "0ZCY0Z80B6"
    },
    {
      "field": "uses",
      "target": "0QEAJB91GZ"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/JPQM16D84G.md",
  "msc": [
    "54A05"
  ],
  "name": "Derived set",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/JPQ/JPQM16D84G.md",
  "status": null,
  "tag": "JPQM16D84G",
  "uri": "m2://JPQM16D84G",
  "url": "https://math2.tamnd.com/wiki/derived-set-definition/",
  "writing": {
    "tag": "JPQM16D84G",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}