---
tag: W1J18NVJYN
kind: statement
role: proposition
name: Topological Vaught conjecture
sources:
  - wikipedia: en:13797188@r1296409660
    locator: Vaught conjecture, section Topological Vaught conjecture
    note: adopted
    mention: wikipedia://en/13797188#65X2
    sections: [Statement]
inventory: [HPFC9BT9Z3]
---

## Statement

The 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.