{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Counterexample\n\nLet $K = \\set{0} \\cup \\set{1/n : n \\ge 1} \\subseteq \\R$.\nThe *comb space* is the subspace of $\\R^2$\n$$C = ([0, 1] \\times \\set{0}) \\cup (K \\times [0, 1]).$$\nIt consists of a base segment, the tooth $\\set{0} \\times [0, 1]$, and the teeth $\\set{1/n} \\times [0, 1]$.\n\nThe space $C$ is [[path-connected|0WB6DH499T]].\nEach point can be joined to $(0, 0)$ by going down its tooth and then along the base.\n\nThe space $C$ is not [[locally connected|FDVAHRJKF3]].\nTake $p = (0, 1)$ and the open set $U = C \\cap B(p, 1/2)$.\nFor large $n$, the set $U$ contains a piece of the tooth $\\set{1/n} \\times [0, 1]$ near height $1$.\nAny open set $V$ with $p \\in V \\subseteq U$ contains such pieces, but $U$ contains no part of the base.\nThe vertical line $x = c$ with $c$ strictly between two teeth splits $V$ into two disjoint open parts, one of them containing $p$.\nSo no open neighbourhood of $p$ inside $U$ is connected.\n\n## Remarks\n\nSo connectedness, even path-connectedness, does not imply local connectedness.\nFor the same reason, $C$ is not [[locally path-connected|1ZVR8BRK65]].\nThe comb space is contractible, so it also shows that contractible spaces need not be locally connected.\n\n## Explanations\n\n### L1 human\n\nPicture a hair comb with infinitely many teeth crowding toward the left edge.\nYou can walk from any point to any other along the teeth and the base, so the comb is in one piece.\nBut near the top of the left edge, every small window shows many separate bits of teeth.\nSo the space is connected overall, yet not connected in the small.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "kind": "counterexample",
    "msc": [
      "54G20",
      "54D05"
    ],
    "name": "Comb space",
    "same_as": {
      "wikidata": "Q5150587",
      "wikipedia": "Comb space"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Comb_space"
      }
    ],
    "tag": "A2Q4RXD162",
    "uses": [
      "0WB6DH499T",
      "FDVAHRJKF3",
      "1ZVR8BRK65"
    ]
  },
  "kind": "counterexample",
  "link": "https://math2.tamnd.com/node/A2Q4RXD162/",
  "links": [
    {
      "field": "uses",
      "target": "0WB6DH499T"
    },
    {
      "field": "uses",
      "target": "FDVAHRJKF3"
    },
    {
      "field": "uses",
      "target": "1ZVR8BRK65"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/A2Q4RXD162.md",
  "msc": [
    "54G20",
    "54D05"
  ],
  "name": "Comb space",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/A2Q/A2Q4RXD162.md",
  "status": null,
  "tag": "A2Q4RXD162",
  "uri": "m2://A2Q4RXD162",
  "url": "https://math2.tamnd.com/wiki/comb-space/",
  "writing": {
    "tag": "A2Q4RXD162",
    "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
      }
    ]
  }
}