{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Statement\n\nThe topological Vaught conjecture is the statement that whenever a Polish group [[acts|0WCY42EV9V]] continuously on a Polish space, there are either countably many [[orbits|0WCY42EV9V]] or continuum many orbits. The topological Vaught conjecture is more general than the original Vaught conjecture: Given a countable language we can form the space of all structures on the [[natural numbers|0MWKJ44V5A]] for that language. If we equip this with the [[topology|0CQHFPGJ45]] generated by first-order formulas, then it is known from A. Gregorczyk, A. Mostowski, C. Ryll-Nardzewski, \"Definability of sets of models of axiomatic theories\" (*Bulletin of the Polish Academy of Sciences (series Mathematics, Astronomy, Physics)*, vol. 9 (1961), pp. 163–7) that the resulting space is Polish. There is a continuous action of the infinite [[symmetric group|0515A6D2Y1]] (the collection of all permutations of the natural numbers with the topology of point-wise convergence) that gives rise to the [[equivalence relation|0G8VEZ3WTK]] of isomorphism. Given a complete first-order theory *T*, the set of structures satisfying *T* is a minimal, [[closed|0G5AV9Q4RR]] invariant set, and hence Polish in its own right.\n",
  "commit": "4a3290d299c8b2cd6493fce2296920ba7f82d942",
  "front": {
    "inventory": [
      "HPFC9BT9Z3"
    ],
    "kind": "statement",
    "name": "Topological Vaught conjecture",
    "role": "proposition",
    "sources": [
      {
        "locator": "Vaught conjecture, section Topological Vaught conjecture",
        "mention": "wikipedia://en/13797188#65X2",
        "note": "adopted",
        "sections": [
          "Statement"
        ],
        "wikipedia": "en:13797188@r1296409660"
      }
    ],
    "tag": "W1J18NVJYN"
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/W1J18NVJYN/",
  "links": null,
  "markdown": "https://math2.tamnd.com/node/W1J18NVJYN.md",
  "msc": null,
  "name": "Topological Vaught conjecture",
  "prov": null,
  "role": "proposition",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/W1J/W1J18NVJYN.md",
  "status": null,
  "tag": "W1J18NVJYN",
  "uri": "m2://W1J18NVJYN",
  "url": "https://math2.tamnd.com/wiki/topological-vaught-conjecture/",
  "writing": {
    "tag": "W1J18NVJYN",
    "kind": "statement",
    "level": "W0",
    "next": [
      {
        "what": "edge uses out",
        "note": "each definition that the statement needs",
        "have": 0,
        "want": 1
      }
    ]
  }
}