{
  "aliases": [
    "real-line-modulo-integers"
  ],
  "backlinks": [
    {
      "field": "uses",
      "target": "03NM7DZXXC"
    }
  ],
  "body": "\n## Example\n\nOn $\\mathbb{R}$, let $s \\sim t$ when $s - t \\in \\mathbb{Z}$. Let $\\mathbb{R} / \\mathbb{Z}$ be the set of classes, with the [[quotient topology|0V8046R0P9]].\nLet $S^1 = \\{ (x, y) \\in \\mathbb{R}^2 : x^2 + y^2 = 1 \\}$ be the unit circle, with the [[subspace topology|0D5YHQGDJY]] of $\\mathbb{R}^2$.\nThen the map $[t] \\mapsto (\\cos 2\\pi t, \\sin 2\\pi t)$ is a [[homeomorphism|05GZ11PWRY]] from $\\mathbb{R} / \\mathbb{Z}$ to $S^1$.\n\n## Remarks\n\nThe map is well defined and continuous by the [[universal property of the quotient topology|0FF2FJ8CH2]], and it is bijective.\nThe space $\\mathbb{R} / \\mathbb{Z}$ is compact, because it is the image of $[0, 1]$ under the quotient map. The circle is Hausdorff. Thus the map is a homeomorphism, by [[this theorem|0430ETCMKD]].\nEquivalently, $S^1$ is homeomorphic to $[0, 1]$ with the two endpoints $0$ and $1$ glued together.\nAddition of classes makes $\\mathbb{R} / \\mathbb{Z}$ a group, the circle group.\n\n## Explanations\n\n### L1 human\n\nWrap the real line around a circle of length $1$. Points that differ by an integer land at the same place.\nThe quotient $\\mathbb{R} / \\mathbb{Z}$ records exactly this wrapping.\n",
  "commit": "6ac761d40018146541484a9e8b9eafd6ecc303dc",
  "front": {
    "aliases": [
      "real-line-modulo-integers"
    ],
    "instance_of": [
      "0CQHFPGJ45"
    ],
    "kind": "example",
    "msc": [
      "54B15"
    ],
    "name": "Circle as a quotient of the real line",
    "sources": [
      {
        "locator": "Periodic interval under addition",
        "url": "https://en.wikipedia.org/wiki/Circle_group"
      },
      {
        "locator": "Topological, measurable, and analytic structure",
        "url": "https://en.wikipedia.org/wiki/Circle_group"
      }
    ],
    "tag": "0WH1GVBJNH",
    "uses": [
      "0V8046R0P9",
      "0FF2FJ8CH2",
      "0430ETCMKD"
    ]
  },
  "kind": "example",
  "link": "https://math2.tamnd.com/node/0WH1GVBJNH/",
  "links": [
    {
      "field": "instance_of",
      "target": "0CQHFPGJ45"
    },
    {
      "field": "uses",
      "target": "0V8046R0P9"
    },
    {
      "field": "uses",
      "target": "0FF2FJ8CH2"
    },
    {
      "field": "uses",
      "target": "0430ETCMKD"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/0WH1GVBJNH.md",
  "msc": [
    "54B15"
  ],
  "name": "Circle as a quotient of the real line",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/0WH/0WH1GVBJNH.md",
  "status": null,
  "tag": "0WH1GVBJNH",
  "uri": "m2://0WH1GVBJNH",
  "url": "https://math2.tamnd.com/wiki/circle-as-a-quotient-of-the-real-line/",
  "writing": {
    "tag": "0WH1GVBJNH",
    "kind": "example",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      }
    ]
  }
}