{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Example\n\nThe *Lakes of Wada* are three disjoint, connected, open subsets of the [[plane|0V3XAMF3C1]] that all have the same [[boundary|05D5EGMFYV]].\nThey can be built from a closed square of land with three lakes of water in it.\nAt step $n$, each lake in turn is extended by a thin canal of open water that winds through the land.\nAfter the step, every point of the remaining land is within distance $2^{-n}$ of each lake.\nThe canals never meet each other, and the land left after each step is a nonempty compact set, so some land remains at the end.\nAfter all steps, each lake is the union of its stages, so it is [[open|02BH3MCBQB]] and [[connected|0V032VWTHD]].\nEvery point of the remaining land is a limit of points of each lake, and the lakes are open and disjoint.\nSo the remaining land is the boundary of each of the three lakes.\n\n## Remarks\n\nThe construction works the same way for any finite number of lakes.\nIn contrast, a simple closed curve in the plane is the common boundary of exactly two regions, by the Jordan curve theorem.\nSo the common boundary of the Lakes of Wada is not a simple closed curve; it is a continuum of a much wilder kind.\nThe example is credited to Takeo Wada; an earlier construction of the same kind is due to L. E. J. Brouwer.\n\n## Explanations\n\n### L1 human\n\nImagine an island with three lakes, and keep digging canals so that the water of each lake comes ever closer to every spot of dry land.\nIn the end, every point of the shore touches all three lakes at once.\nIt sounds impossible, because on a map two countries usually meet along a line and three meet only at points.\nThe example shows that boundaries in the plane can be far wilder than drawn lines.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "illustrates": [
      "05D5EGMFYV"
    ],
    "kind": "example",
    "msc": [
      "54F15",
      "54G15"
    ],
    "name": "Lakes of Wada",
    "same_as": {
      "wikidata": "Q1331442",
      "wikipedia": "Lakes of Wada"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Lakes_of_Wada"
      }
    ],
    "tag": "3ENQE05H1H",
    "uses": [
      "0V3XAMF3C1",
      "02BH3MCBQB",
      "0V032VWTHD"
    ]
  },
  "kind": "example",
  "link": "https://math2.tamnd.com/node/3ENQE05H1H/",
  "links": [
    {
      "field": "illustrates",
      "target": "05D5EGMFYV"
    },
    {
      "field": "uses",
      "target": "0V3XAMF3C1"
    },
    {
      "field": "uses",
      "target": "02BH3MCBQB"
    },
    {
      "field": "uses",
      "target": "0V032VWTHD"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/3ENQE05H1H.md",
  "msc": [
    "54F15",
    "54G15"
  ],
  "name": "Lakes of Wada",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/3EN/3ENQE05H1H.md",
  "status": null,
  "tag": "3ENQE05H1H",
  "uri": "m2://3ENQE05H1H",
  "url": "https://math2.tamnd.com/wiki/lakes-of-wada/",
  "writing": {
    "tag": "3ENQE05H1H",
    "kind": "example",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}