{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Counterexample\n\nThe Alexander horned sphere is a subset $\\Sigma \\subset \\R^3$ built by an infinite process.\nIt is the image of a [[topological embedding|00ZG9Y9TR4]] of the $2$-[[sphere|T62369RFH6]] $S^2$.\nStart with a ball and pull out two horns that grow towards each other.\nNear their tips, let each horn split into a pair of thinner horns.\nLet the two pairs hook through each other like two links of a chain, without touching.\nRepeat this splitting inside each pair, at smaller and smaller scales, infinitely often.\nThe limit set of the surfaces is $\\Sigma$; the tips shrink to a Cantor set of points, and $\\Sigma$ is still homeomorphic to $S^2$.\n\nThe closure of the bounded component of $\\R^3 \\setminus \\Sigma$ is homeomorphic to the closed ball $D^3$.\nThe unbounded component is not simply connected.\nA loop around the base of one horn cannot be shrunk to a point in the complement.\nEach attempt to slide it off the horns is blocked by the next pair of hooked horns.\nSo the closure of the unbounded component is not homeomorphic to the closure of the complement of a round ball, whose [[fundamental group|S1XK3HXHFX]] is trivial.\n\n## Remarks\n\nThe example shows that the [[Schoenflies theorem|3VX06XRVZG]] fails in dimension $3$ without its local flatness hypothesis.\nThe sphere $\\Sigma$ is not locally flat at the points of the Cantor set.\nJ. W. Alexander described it in 1924.\nGluing two copies of the closed bounded side along $\\Sigma$ gives a space homeomorphic to $S^3$, a result of R. H. Bing.\n\n## Explanations\n\n### L1 human\n\nIn the plane, every loop bounds a disk on the inside and the rest of the plane on the outside, both as nice as possible.\nIn space one would expect the same for a sphere, but the horned sphere breaks this.\nIts horns branch and interlock forever, so a rubber band around a horn on the outside can never be pulled free.\nThe surface is still a perfect topological sphere; the problem is only how it sits in space.\n",
  "commit": "5fc425f97f1f6b11da5892c2948ef8c5d8805665",
  "front": {
    "kind": "counterexample",
    "msc": [
      "57M30",
      "57N45"
    ],
    "name": "Alexander horned sphere",
    "same_as": {
      "wikidata": "Q1361755",
      "wikipedia": "Alexander horned sphere"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Alexander_horned_sphere"
      }
    ],
    "tag": "QQEASX79RW",
    "uses": [
      "3VX06XRVZG",
      "T62369RFH6",
      "00ZG9Y9TR4",
      "S1XK3HXHFX"
    ]
  },
  "kind": "counterexample",
  "link": "https://math2.tamnd.com/node/QQEASX79RW/",
  "links": [
    {
      "field": "uses",
      "target": "3VX06XRVZG"
    },
    {
      "field": "uses",
      "target": "T62369RFH6"
    },
    {
      "field": "uses",
      "target": "00ZG9Y9TR4"
    },
    {
      "field": "uses",
      "target": "S1XK3HXHFX"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/QQEASX79RW.md",
  "msc": [
    "57M30",
    "57N45"
  ],
  "name": "Alexander horned sphere",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/QQE/QQEASX79RW.md",
  "status": null,
  "tag": "QQEASX79RW",
  "uri": "m2://QQEASX79RW",
  "url": "https://math2.tamnd.com/wiki/alexander-horned-sphere/",
  "writing": {
    "tag": "QQEASX79RW",
    "kind": "counterexample",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      },
      {
        "what": "edge refutes or not_instance_of out",
        "have": 0,
        "want": 1
      }
    ]
  }
}