{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nA *totally disconnected group* is a [[topological group|92700QAB5Y]] $G$ whose underlying topological space is [[totally disconnected|0RJSBR9GW1]].\nThat is, the only connected subsets of $G$ are the empty set and the one-point sets.\n\n## Remarks\n\nExample: the [[p-adic integers|V6TJAFZ4MD]] $\\Z_p$ under addition.\nEach subgroup $p^n \\Z_p$ is open and closed, and two distinct elements $x, y$ lie in different cosets of $p^n \\Z_p$ for $n$ large, so no connected set contains both.\nNon-example: $(\\R, +)$ with the usual topology is connected, so it is not totally disconnected.\nEquivalently, $G$ is totally disconnected if and only if its identity component is $\\set{e}$, because each connected component is a coset of the identity component.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "AW3J9AQSA0",
    "kind": "definition",
    "msc": [
      "22A05",
      "54H11"
    ],
    "name": "Totally disconnected group",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Totally_disconnected_group"
      }
    ],
    "tag": "CZG6Y6EAHP",
    "uses": [
      "92700QAB5Y",
      "0RJSBR9GW1",
      "V6TJAFZ4MD"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/CZG6Y6EAHP/",
  "links": [
    {
      "field": "defines",
      "target": "AW3J9AQSA0"
    },
    {
      "field": "uses",
      "target": "92700QAB5Y"
    },
    {
      "field": "uses",
      "target": "0RJSBR9GW1"
    },
    {
      "field": "uses",
      "target": "V6TJAFZ4MD"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/CZG6Y6EAHP.md",
  "msc": [
    "22A05",
    "54H11"
  ],
  "name": "Totally disconnected group",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/CZG/CZG6Y6EAHP.md",
  "status": null,
  "tag": "CZG6Y6EAHP",
  "uri": "m2://CZG6Y6EAHP",
  "url": "https://tamnd.github.io/math2/wiki/totally-disconnected-group-definition/",
  "writing": {
    "tag": "CZG6Y6EAHP",
    "kind": "definition",
    "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
      }
    ]
  }
}