{
  "aliases": null,
  "backlinks": [
    {
      "field": "defines",
      "target": "8M0QH4Z60S"
    }
  ],
  "body": "\n## Summary\n\nA regular open set in a [[topological space|0CQHFPGJ45]] is an [[open set|02BH3MCBQB]] that equals the [[interior|0RZH6W98HK]] of its [[closure|0QEAJB91GZ]].\nSuch a set has no \"cracks\": it does not miss single points or thin pieces inside its own closure.\nThe regular open sets of any space form a complete [[Boolean algebra|0XADJKM699]], which makes them useful in set theory and in pointless topology.\n\n## Explanations\n\n### L1 human\n\nTake an open set, fill in its boundary by taking the closure, and then remove the boundary again by taking the interior.\nIf you get back the same set, the set is regular open.\nAn open interval $(0, 1)$ in $\\R$ is regular open: closing gives $[0, 1]$, and the interior of that is $(0, 1)$ again.\nThe set $(0, 1) \\cup (1, 2)$ is not, because the process fills the gap at $1$ and returns $(0, 2)$.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "kind": "concept",
    "msc": [
      "54A05"
    ],
    "name": "Regular open set",
    "same_as": {
      "wikidata": "Q110745661",
      "wikipedia": "Regular open set"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Regular_open_set"
      }
    ],
    "specializes": [
      "02BH3MCBQB"
    ],
    "tag": "833XAA2Q65"
  },
  "kind": "concept",
  "link": "https://math2.tamnd.com/node/833XAA2Q65/",
  "links": [
    {
      "field": "specializes",
      "target": "02BH3MCBQB"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/833XAA2Q65.md",
  "msc": [
    "54A05"
  ],
  "name": "Regular open set",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/833/833XAA2Q65.md",
  "status": null,
  "tag": "833XAA2Q65",
  "uri": "m2://833XAA2Q65",
  "url": "https://math2.tamnd.com/wiki/regular-open-set/",
  "writing": {
    "tag": "833XAA2Q65",
    "kind": "concept",
    "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
      }
    ]
  }
}