{
  "aliases": null,
  "backlinks": [
    {
      "field": "defines",
      "target": "9SEF1CWT6X"
    },
    {
      "field": "specializes",
      "target": "RA56GHB9JZ"
    }
  ],
  "body": "\n## Summary\n\nA topological semigroup is a [[semigroup|0XMEM406JM]] with a topology in which the semigroup operation is jointly continuous.\nIt is a very general setting where algebra and topology meet through one associative operation.\nTopological monoids and topological groups are special cases.\nCompact topological semigroups have a rich structure theory with uses in topological dynamics.\n\n## Explanations\n\n### L1 human\n\nA semigroup is a set with an associative way of combining two elements, without any demand for an identity or inverses.\nA topological semigroup adds a notion of nearness, and asks that combining nearby elements gives nearby results.\nFor example, the interval $[0, 1]$ with ordinary multiplication is a topological semigroup.\nThe product of two numbers in $[0, 1]$ is again in $[0, 1]$, and the product depends continuously on both factors.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "22A15"
    ],
    "name": "Topological semigroup",
    "same_as": {
      "wikidata": "Q7825043",
      "wikipedia": "Topological semigroup"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Topological_semigroup"
      }
    ],
    "specializes": [
      "0XMEM406JM"
    ],
    "tag": "QXB6JDG68F"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/QXB6JDG68F/",
  "links": [
    {
      "field": "specializes",
      "target": "0XMEM406JM"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/QXB6JDG68F.md",
  "msc": [
    "22A15"
  ],
  "name": "Topological semigroup",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/QXB/QXB6JDG68F.md",
  "status": null,
  "tag": "QXB6JDG68F",
  "uri": "m2://QXB6JDG68F",
  "url": "https://tamnd.github.io/math2/wiki/topological-semigroup/",
  "writing": {
    "tag": "QXB6JDG68F",
    "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
      }
    ]
  }
}