{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Proof\n\nWe prove the two directions separately.\n\n1. Let $A$ be [[open|01ZQQDCERP]] and let $x \\in A$. Take $U = A$. Then $U$ is open and $x \\in U \\subseteq A$. So $A$ is a neighbourhood of $x$.\n2. Conversely, let $A$ be a neighbourhood of each of its points. For each $x \\in A$, choose an open set $U_x$ with $x \\in U_x \\subseteq A$.\n3. Let $W = \\bigcup_{x \\in A} U_x$, the [[union|05GK32H72H]] of this family. Each $U_x \\subseteq A$, so $W \\subseteq A$.\n4. Each $x \\in A$ lies in $U_x$, so $A \\subseteq W$. Hence $A = W$.\n5. By the axioms of a [[topology|0CNQXVBHCX]], a union of members of $\\tau$ is in $\\tau$. This includes the empty family, whose union is $\\varnothing$. So $A = W$ is open.\n\n## Idea\n\nAn open set is its own open neighbourhood at each point. Conversely, if each point has an open set around it inside $A$, then $A$ is the union of these open sets, and unions of open sets are open.\n",
  "commit": "4a3290d299c8b2cd6493fce2296920ba7f82d942",
  "front": {
    "kind": "proof",
    "msc": [
      "54A05"
    ],
    "name": "Proof that open sets are the sets that are neighbourhoods of all their points",
    "proves": "09CJ00H3NW",
    "sources": [
      {
        "locator": "Neighbourhood of a point",
        "url": "https://en.wikipedia.org/wiki/Neighbourhood_(mathematics)"
      }
    ],
    "tag": "3MH8WKGV83",
    "uses": [
      "0CNQXVBHCX",
      "01ZQQDCERP",
      "05GK32H72H"
    ]
  },
  "kind": "proof",
  "link": "https://math2.tamnd.com/node/3MH8WKGV83/",
  "links": [
    {
      "field": "proves",
      "target": "09CJ00H3NW"
    },
    {
      "field": "uses",
      "target": "0CNQXVBHCX"
    },
    {
      "field": "uses",
      "target": "01ZQQDCERP"
    },
    {
      "field": "uses",
      "target": "05GK32H72H"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/3MH8WKGV83.md",
  "msc": [
    "54A05"
  ],
  "name": "Proof that open sets are the sets that are neighbourhoods of all their points",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/3MH/3MH8WKGV83.md",
  "status": null,
  "tag": "3MH8WKGV83",
  "uri": "m2://3MH8WKGV83",
  "url": "https://math2.tamnd.com/wiki/proof-that-open-sets-are-the-sets-that-are-neighbourhoods-of-all-their-points/",
  "writing": {
    "tag": "3MH8WKGV83",
    "kind": "proof",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      }
    ]
  }
}