Vaught Conjecture - Topological Vaught Conjecture

Topological Vaught Conjecture

The topological Vaught conjecture is the statement that whenever a Polish group acts continuously on a Polish space, there are either countably many orbits 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 for that language. If we equip this with the topology 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 (the collection of all permutations of the natural numbers with the topology of point wise convergence) which gives rise to the equivalence relation of isomorphism. Given a complete first order theory T, the set of structures satisfying T is a minimal, closed invariant set, and hence Polish in its own right.