{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Counterexample\n\nLet $X$ be an infinite set and let $p \\in X$.\nDeclare $U \\subseteq X$ open if $p \\notin U$, or if $p \\in U$ and $X \\setminus U$ is finite.\nThis is the Fort space on $X$ with point $p$.\nIt is [[compact|0TVJZYB6SH]] and [[Hausdorff|0S1FJ8VEX9]].\nIf $X$ is [[uncountable|0W7FBYS1F1]], it is not [[first-countable|0CP5NDD21C]], and so not [[metrizable|091GJ3FYD7]].\n\n## Remarks\n\nCompactness: in an open cover, one member $U$ contains $p$, and the finitely many points outside $U$ need finitely many more members.\nHausdorff: two points $x, y \\ne p$ are separated by $\\set{x}$ and $\\set{y}$, and $x \\ne p$ is separated from $p$ by $\\set{x}$ and $X \\setminus \\set{x}$.\nLet $X$ be uncountable and let $U_1, U_2, \\dots$ be open sets containing $p$.\nTheir intersection has countable complement, so it contains a point $x \\ne p$.\nThen $X \\setminus \\set{x}$ is an open set containing $p$ that contains no $U_n$, so $p$ has no countable neighbourhood base.\nThe Fort space is the [[one-point compactification|0GWN3CMRK8]] of the discrete space $X \\setminus \\set{p}$.\nIf $X$ is countably infinite, it is homeomorphic to $\\set{0} \\cup \\set{1/n : n \\ge 1}$ with the usual topology, which is metrizable.\n\n## Explanations\n\n### L1 human\n\nTake a set of isolated points and add one extra point $p$.\nA set around $p$ is open when it leaves out only finitely many points.\nThis makes the space compact and keeps points apart.\nWith uncountably many points, no countable list of open sets can pin down $p$, so no distance function gives this topology.\n",
  "commit": "5fc425f97f1f6b11da5892c2948ef8c5d8805665",
  "front": {
    "kind": "counterexample",
    "msc": [
      "54G20",
      "54D30"
    ],
    "name": "Fort space",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Fort_space"
      }
    ],
    "tag": "7FYKA5F3S6",
    "uses": [
      "0TVJZYB6SH",
      "0S1FJ8VEX9",
      "0CP5NDD21C",
      "091GJ3FYD7",
      "0GWN3CMRK8",
      "0W7FBYS1F1"
    ]
  },
  "kind": "counterexample",
  "link": "https://math2.tamnd.com/node/7FYKA5F3S6/",
  "links": [
    {
      "field": "uses",
      "target": "0TVJZYB6SH"
    },
    {
      "field": "uses",
      "target": "0S1FJ8VEX9"
    },
    {
      "field": "uses",
      "target": "0CP5NDD21C"
    },
    {
      "field": "uses",
      "target": "091GJ3FYD7"
    },
    {
      "field": "uses",
      "target": "0GWN3CMRK8"
    },
    {
      "field": "uses",
      "target": "0W7FBYS1F1"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/7FYKA5F3S6.md",
  "msc": [
    "54G20",
    "54D30"
  ],
  "name": "Fort space",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/7FY/7FYKA5F3S6.md",
  "status": null,
  "tag": "7FYKA5F3S6",
  "uri": "m2://7FYKA5F3S6",
  "url": "https://math2.tamnd.com/wiki/fort-space/",
  "writing": {
    "tag": "7FYKA5F3S6",
    "kind": "counterexample",
    "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
      },
      {
        "what": "edge refutes or not_instance_of out",
        "have": 0,
        "want": 1
      }
    ]
  }
}