{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $M$ be a [[topological space|0CQHFPGJ45]] with a binary operation $M \\times M \\to M$, written $(x, y) \\mapsto xy$.\nThe operation is a *continuous multiplication* if the map $\\mu \\colon M \\times M \\to M$, $\\mu(x, y) = xy$, is [[continuous|0VGFHTM75Y]].\nHere $M \\times M$ carries the [[product topology|0GBG8B49AA]].\n\n## Remarks\n\nExample: $\\R$ with its usual topology and ordinary multiplication.\nThe map $(x, y) \\mapsto xy$ is a polynomial in the coordinates, so it is continuous on $\\R^2$.\nNon-example: $\\Z$ with the cofinite topology and addition.\nEvery nonempty open set $U \\times V$ of $\\Z \\times \\Z$ meets the set $\\set{(x, -x) : x \\in \\Z}$, because $U$ and $-V$ are cofinite and so they intersect.\nSo the preimage of the open set $\\Z \\setminus \\set{0}$ under addition contains no nonempty open set, and addition is not continuous.\nContinuity of $\\mu$ implies that every left translation $y \\mapsto xy$ and every right translation $x \\mapsto xy$ is continuous.\nThe converse fails: in the non-example above every translation is a bijection of $\\Z$, hence a homeomorphism for the cofinite topology.\nIn additive notation the same condition is called continuous addition.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "ZZK2GM1V15",
    "kind": "definition",
    "msc": [
      "22A15",
      "22A05"
    ],
    "name": "Continuous multiplication",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Topological_group"
      }
    ],
    "tag": "74TDTC7ETT",
    "uses": [
      "0CQHFPGJ45",
      "0GBG8B49AA",
      "0VGFHTM75Y"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/74TDTC7ETT/",
  "links": [
    {
      "field": "defines",
      "target": "ZZK2GM1V15"
    },
    {
      "field": "uses",
      "target": "0CQHFPGJ45"
    },
    {
      "field": "uses",
      "target": "0GBG8B49AA"
    },
    {
      "field": "uses",
      "target": "0VGFHTM75Y"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/74TDTC7ETT.md",
  "msc": [
    "22A15",
    "22A05"
  ],
  "name": "Continuous multiplication",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/74T/74TDTC7ETT.md",
  "status": null,
  "tag": "74TDTC7ETT",
  "uri": "m2://74TDTC7ETT",
  "url": "https://tamnd.github.io/math2/wiki/continuous-multiplication-definition/",
  "writing": {
    "tag": "74TDTC7ETT",
    "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
      }
    ]
  }
}