{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Example\n\nLet $\\sim$ be the equivalence relation on the square $[0, 1] \\times [0, 1]$ generated by $(0, y) \\sim (1, y)$ and $(x, 0) \\sim (x, 1)$ for all $x, y \\in [0, 1]$.\nThe quotient space $[0, 1]^2 / {\\sim}$, with the [[quotient topology|0V8046R0P9]], is the torus $T^2$.\n\n## Remarks\n\nThe torus is homeomorphic to the product $S^1 \\times S^1$ of two [[circles|0WH1GVBJNH]], with the [[product topology|0Z57EC6YX3]], and to $\\mathbb{R}^2 / \\mathbb{Z}^2$.\nIt is also homeomorphic to a surface in $\\mathbb{R}^3$: turn a circle about an axis in its plane that does not meet the circle.\nThe torus is a compact, connected surface. It is not homeomorphic to the sphere.\n\n## Explanations\n\n### L1 human\n\nTake a square sheet. Glue the left edge to the right edge to get a tube, and then glue the two end circles of the tube.\nThe result is the surface of a doughnut.\n",
  "commit": "f591989ed15704e9d8755a7a5a6426d73de7a3ae",
  "front": {
    "instance_of": [
      "0CQHFPGJ45"
    ],
    "kind": "example",
    "msc": [
      "57K20"
    ],
    "name": "Torus",
    "same_as": {
      "wikidata": "Q12510",
      "wikipedia": "Torus"
    },
    "sources": [
      {
        "locator": "Topology",
        "url": "https://en.wikipedia.org/wiki/Torus"
      }
    ],
    "tag": "03NM7DZXXC",
    "uses": [
      "0V8046R0P9",
      "0Z57EC6YX3",
      "0WH1GVBJNH"
    ]
  },
  "kind": "example",
  "link": "https://math2.tamnd.com/node/03NM7DZXXC/",
  "links": [
    {
      "field": "instance_of",
      "target": "0CQHFPGJ45"
    },
    {
      "field": "uses",
      "target": "0V8046R0P9"
    },
    {
      "field": "uses",
      "target": "0Z57EC6YX3"
    },
    {
      "field": "uses",
      "target": "0WH1GVBJNH"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/03NM7DZXXC.md",
  "msc": [
    "57K20"
  ],
  "name": "Torus",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/03N/03NM7DZXXC.md",
  "status": null,
  "tag": "03NM7DZXXC",
  "uri": "m2://03NM7DZXXC",
  "url": "https://math2.tamnd.com/wiki/torus/",
  "writing": {
    "tag": "03NM7DZXXC",
    "kind": "example",
    "level": "W2",
    "next": [
      {
        "what": "a person confirmed each agent-written section",
        "note": "rule P04",
        "have": 0,
        "want": 1
      }
    ]
  }
}