{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Proof\n\n1. Let $(X, d)$ be a [[metric space|063W5JMY85]], let $I$ be a nonempty set, and let $(F_i)_{i \\in I}$ be a family of [[closed subsets|0G5AV9Q4RR]] of $X$.\n2. Put $F = \\bigcap_{i \\in I} F_i$, the [[intersection|0839XNQXDK]] of the family.\n3. We claim that $X \\setminus F = \\bigcup_{i \\in I} (X \\setminus F_i)$, a [[union|05GK32H72H]] of [[complements|0B1FSS1VZW]].\n4. Let $x \\in X$. Then $x \\notin F$ if and only if $x \\notin F_j$ for some $j \\in I$.\n5. This holds if and only if $x \\in X \\setminus F_j$ for some $j \\in I$, that is, $x \\in \\bigcup_{i \\in I} (X \\setminus F_i)$. This proves the claim.\n6. Each $F_i$ is closed, so each $X \\setminus F_i$ is open, by the definition of a closed set.\n7. By [[union of open sets is open|0HEF46RPEK]], the set $\\bigcup_{i \\in I} (X \\setminus F_i)$ is open. By step 3, $X \\setminus F$ is open.\n8. By the definition of a closed set, $F$ is closed in $X$.\n\n## Idea\n\nComplements turn intersections into unions. A union of open sets is open, so the complement of the intersection is open. That is what it means for the intersection to be closed.\n",
  "commit": "68a8c3dc73fd55004b2796431c607c30960d1a85",
  "front": {
    "kind": "proof",
    "msc": [
      "54E35"
    ],
    "name": "Proof that an intersection of closed sets of a metric space is closed",
    "proves": "09YMV8H4X5",
    "sources": [
      {
        "url": "https://proofwiki.org/wiki/Intersection_of_Closed_Sets_is_Closed_in_Topological_Space"
      }
    ],
    "tag": "870ABH4GMQ",
    "uses": [
      "063W5JMY85",
      "0G5AV9Q4RR",
      "0839XNQXDK",
      "05GK32H72H",
      "0B1FSS1VZW",
      "0HEF46RPEK"
    ]
  },
  "kind": "proof",
  "link": "https://math2.tamnd.com/node/870ABH4GMQ/",
  "links": [
    {
      "field": "proves",
      "target": "09YMV8H4X5"
    },
    {
      "field": "uses",
      "target": "063W5JMY85"
    },
    {
      "field": "uses",
      "target": "0G5AV9Q4RR"
    },
    {
      "field": "uses",
      "target": "0839XNQXDK"
    },
    {
      "field": "uses",
      "target": "05GK32H72H"
    },
    {
      "field": "uses",
      "target": "0B1FSS1VZW"
    },
    {
      "field": "uses",
      "target": "0HEF46RPEK"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/870ABH4GMQ.md",
  "msc": [
    "54E35"
  ],
  "name": "Proof that an intersection of closed sets of a metric space is closed",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/870/870ABH4GMQ.md",
  "status": null,
  "tag": "870ABH4GMQ",
  "uri": "m2://870ABH4GMQ",
  "url": "https://math2.tamnd.com/wiki/proof-that-an-intersection-of-closed-sets-of-a-metric-space-is-closed/",
  "writing": {
    "tag": "870ABH4GMQ",
    "kind": "proof",
    "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
      }
    ]
  }
}