{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "00BWV5CKCF"
    },
    {
      "field": "uses",
      "target": "066HW6D9A7"
    },
    {
      "field": "uses",
      "target": "09ZKE5XQ74"
    },
    {
      "field": "uses",
      "target": "0K9AAJ8G51"
    },
    {
      "field": "uses",
      "target": "0MYFJVMN7M"
    },
    {
      "field": "uses",
      "target": "0R7BTEWCJ2"
    }
  ],
  "body": "\n## Statement\n\nLet $X$ be a [[topological space|0CNQXVBHCX]] and let $K \\subseteq X$.\nThen $K$ is [[compact|00R0X1WGPX]] if and only if each family $(U_i)_{i \\in I}$ of open subsets of $X$ with $K \\subseteq \\bigcup_{i \\in I} U_i$ has a finite subfamily $U_{i_1}, \\ldots, U_{i_n}$ with $K \\subseteq U_{i_1} \\cup \\cdots \\cup U_{i_n}$.\n\n## Remarks\n\nProof: the open sets of the [[subspace topology|0D5YHQGDJY]] on $K$ are the sets $U \\cap K$ with $U$ open in $X$. So covers of $K$ by open sets of $K$ and covers by open sets of $X$ match each other.\nFor $K = X$, the statement is the definition of a [[compact space|0QPHW703PV]].\nThus compactness of a subset does not depend on the space around it. If $K \\subseteq Y \\subseteq X$, then $K$ is compact in $Y$ if and only if it is compact in $X$.\n\n## Explanations\n\n### L1 human\n\nTo test if a subset is compact, you can use open sets of the big space.\nEach time you cover the subset with open sets, finitely many of them must already be enough.\n",
  "commit": "6ac761d40018146541484a9e8b9eafd6ecc303dc",
  "front": {
    "kind": "statement",
    "msc": [
      "54D30"
    ],
    "name": "Compactness via open covers",
    "role": "proposition",
    "sources": [
      {
        "locator": "Compactness of subsets",
        "url": "https://en.wikipedia.org/wiki/Compact_space"
      }
    ],
    "tag": "0P6PMJB4MA",
    "uses": [
      "00R0X1WGPX",
      "0NVPNC12JY",
      "0D5YHQGDJY"
    ]
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/0P6PMJB4MA/",
  "links": [
    {
      "field": "uses",
      "target": "00R0X1WGPX"
    },
    {
      "field": "uses",
      "target": "0NVPNC12JY"
    },
    {
      "field": "uses",
      "target": "0D5YHQGDJY"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/0P6PMJB4MA.md",
  "msc": [
    "54D30"
  ],
  "name": "Compactness via open covers",
  "prov": null,
  "role": "proposition",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/0P6/0P6PMJB4MA.md",
  "status": null,
  "tag": "0P6PMJB4MA",
  "uri": "m2://0P6PMJB4MA",
  "url": "https://math2.tamnd.com/wiki/compactness-via-open-covers/",
  "writing": {
    "tag": "0P6PMJB4MA",
    "kind": "statement",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      }
    ]
  }
}