{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $X$ be a [[topological space|0CQHFPGJ45]].\nFor $S \\subseteq X$, write $\\overline{S}$ for the [[closure|0QEAJB91GZ]] of $S$ and $\\operatorname{int} S$ for its [[interior|0RZH6W98HK]].\nA subset $U \\subseteq X$ is *regular open* if\n$$U = \\operatorname{int} \\overline{U}.$$\n\n## Remarks\n\nA regular open set is [[open|02BH3MCBQB]], since an interior is always open.\nFor every open $U$, one has $U \\subseteq \\operatorname{int} \\overline{U}$, and the set $\\operatorname{int} \\overline{U}$ is regular open.\n\nExample: in $\\R$, let $U = (0, 1)$.\nThen $\\overline{U} = [0, 1]$ and $\\operatorname{int} [0, 1] = (0, 1) = U$, so $U$ is regular open.\n\nNon-example: in $\\R$, let $U = (0, 1) \\cup (1, 2)$.\nThen $\\overline{U} = [0, 2]$ and $\\operatorname{int} \\overline{U} = (0, 2) \\ne U$, so $U$ is open but not regular open.\n\nDually, a set $F$ is *regular closed* if $F = \\overline{\\operatorname{int} F}$.\nA set is regular open if and only if its complement is regular closed.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "defines": "833XAA2Q65",
    "kind": "definition",
    "msc": [
      "54A05"
    ],
    "name": "Regular open set",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Regular_open_set"
      }
    ],
    "tag": "8M0QH4Z60S",
    "uses": [
      "0CQHFPGJ45",
      "02BH3MCBQB",
      "0QEAJB91GZ",
      "0RZH6W98HK"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/8M0QH4Z60S/",
  "links": [
    {
      "field": "defines",
      "target": "833XAA2Q65"
    },
    {
      "field": "uses",
      "target": "0CQHFPGJ45"
    },
    {
      "field": "uses",
      "target": "02BH3MCBQB"
    },
    {
      "field": "uses",
      "target": "0QEAJB91GZ"
    },
    {
      "field": "uses",
      "target": "0RZH6W98HK"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/8M0QH4Z60S.md",
  "msc": [
    "54A05"
  ],
  "name": "Regular open set",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/8M0/8M0QH4Z60S.md",
  "status": null,
  "tag": "8M0QH4Z60S",
  "uri": "m2://8M0QH4Z60S",
  "url": "https://math2.tamnd.com/wiki/regular-open-set-definition/",
  "writing": {
    "tag": "8M0QH4Z60S",
    "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
      }
    ]
  }
}