{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "428R02EYWS"
    },
    {
      "field": "uses",
      "target": "FPMJEX3DSJ"
    },
    {
      "field": "uses",
      "target": "N1WJR10WX0"
    },
    {
      "field": "uses",
      "target": "X057MN9YY9"
    },
    {
      "field": "uses",
      "target": "ZYCX4PN96G"
    },
    {
      "field": "defines",
      "target": "GDTQD3J2RH"
    }
  ],
  "body": "\n## Summary\n\nA universal cover of a connected space $X$ is a [[covering space|3XZRRE165N]] of $X$ whose total space is simply connected.\nIt exists for every connected, locally path-connected and [[semilocally simply connected|R3SHZX0NZR]] space, and it is unique up to isomorphism.\nIt covers every other connected covering space of $X$, and its group of deck transformations is isomorphic to the [[fundamental group|S1XK3HXHFX]] of $X$.\n\n## Explanations\n\n### L1 human\n\nThe universal cover unwraps every loop of a space that cannot be shrunk.\nThe real line wound around the circle as a helix is the universal cover of the circle.\nGoing once around the circle moves one step up the helix, so no loop survives upstairs.\nThe plane is the universal cover of the torus, through the map that reduces both coordinates modulo $1$.\nThe universal cover is the largest connected covering, and every other connected covering sits between it and the space.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "kind": "concept",
    "msc": [
      "57M10",
      "55R05"
    ],
    "name": "Universal cover",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Covering_space"
      }
    ],
    "specializes": [
      "3XZRRE165N"
    ],
    "tag": "Y1YPE6PM52"
  },
  "kind": "concept",
  "link": "https://math2.tamnd.com/node/Y1YPE6PM52/",
  "links": [
    {
      "field": "specializes",
      "target": "3XZRRE165N"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/Y1YPE6PM52.md",
  "msc": [
    "57M10",
    "55R05"
  ],
  "name": "Universal cover",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/Y1Y/Y1YPE6PM52.md",
  "status": null,
  "tag": "Y1YPE6PM52",
  "uri": "m2://Y1YPE6PM52",
  "url": "https://math2.tamnd.com/wiki/universal-cover/",
  "writing": {
    "tag": "Y1YPE6PM52",
    "kind": "concept",
    "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
      }
    ]
  }
}
