{
  "aliases": null,
  "backlinks": [
    {
      "field": "defines",
      "target": "4XR88RPJ4K"
    }
  ],
  "body": "\n## Summary\n\nThe Bohr compactification of a topological group $G$ is the largest compact Hausdorff group into which $G$ maps continuously.\nEvery continuous homomorphism from $G$ to a compact Hausdorff group factors through it in a unique way.\nFor a locally compact abelian group it is the dual of the dual group made discrete.\nThe continuous functions on it correspond to the almost periodic functions on $G$.\n\n## Explanations\n\n### L1 human\n\nMany groups, like the real line, are not compact, but compact groups are easier to analyse.\nThe Bohr compactification is the best compact group that \"sees\" $G$: it keeps exactly the information that compact groups can detect.\nFor the real line, it keeps all the waves $x \\mapsto e^{i\\xi x}$ but forgets the topology of the frequencies $\\xi$.\nThe resulting compact group is huge, but the continuous functions on it correspond exactly to the almost periodic functions studied by Harald Bohr, such as $\\cos x + \\cos(\\sqrt{2}x)$.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "22C05",
      "43A60",
      "22B05"
    ],
    "name": "Bohr compactification",
    "same_as": {
      "wikidata": "Q4938217",
      "wikipedia": "Bohr compactification"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Bohr_compactification"
      }
    ],
    "tag": "G852FHZ3W1"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/G852FHZ3W1/",
  "links": null,
  "markdown": "https://tamnd.github.io/math2/node/G852FHZ3W1.md",
  "msc": [
    "22C05",
    "43A60",
    "22B05"
  ],
  "name": "Bohr compactification",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/G85/G852FHZ3W1.md",
  "status": null,
  "tag": "G852FHZ3W1",
  "uri": "m2://G852FHZ3W1",
  "url": "https://tamnd.github.io/math2/wiki/bohr-compactification/",
  "writing": {
    "tag": "G852FHZ3W1",
    "kind": "concept",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}