{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "09YMV8H4X5"
    }
  ],
  "body": "\n## Statement\n\nLet $(X, d)$ be a [[metric space|063W5JMY85]] and let $(U_i)_{i \\in I}$ be a family of [[open subsets|01ZQQDCERP]] of $X$, indexed by any set $I$.\nThen the [[union|05GK32H72H]] $\\bigcup_{i \\in I} U_i$ is open in $X$.\n\n## Remarks\n\nProof: let $x$ be in the union. Then $x \\in U_j$ for some $j \\in I$, and there is $\\varepsilon \u003e 0$ with $B(x, \\varepsilon) \\subseteq U_j$. This ball is also in the union. Then use [[open sets of a metric space|0A26X79RTB]].\nThe family can be infinite, and even uncountable.\nIn a general [[topological space|0CNQXVBHCX]], this property is one of the axioms.\n\n## Explanations\n\n### L1 human\n\nEach point of the union lies in one of the open sets, and that set already gives a small ball about the point.\nFor example, the union of the intervals $(1/n, 1)$ for $n \\ge 2$ is the open interval $(0, 1)$.\n",
  "commit": "545acf83a9b75c8ddc375ef8b23d2ae101a4d7e4",
  "front": {
    "kind": "statement",
    "msc": [
      "54E35"
    ],
    "name": "Union of open sets is open",
    "role": "proposition",
    "sources": [
      {
        "locator": "Properties",
        "url": "https://en.wikipedia.org/wiki/Open_set"
      }
    ],
    "tag": "0HEF46RPEK",
    "uses": [
      "0A26X79RTB",
      "05GK32H72H"
    ]
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/0HEF46RPEK/",
  "links": [
    {
      "field": "uses",
      "target": "0A26X79RTB"
    },
    {
      "field": "uses",
      "target": "05GK32H72H"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/0HEF46RPEK.md",
  "msc": [
    "54E35"
  ],
  "name": "Union of open sets is open",
  "prov": null,
  "role": "proposition",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/0HE/0HEF46RPEK.md",
  "status": null,
  "tag": "0HEF46RPEK",
  "uri": "m2://0HEF46RPEK",
  "url": "https://math2.tamnd.com/wiki/union-of-open-sets-is-open/",
  "writing": {
    "tag": "0HEF46RPEK",
    "kind": "statement",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      }
    ]
  }
}