Vaught’s conjecture on analytic sets

In each of these case the result was shortly or immediately after extended to analytic sets. For this purpose let us write TVC(G,Σ∼ 1 1) if whenever G acts continuously on a Polish space X and A ⊂ X is Σ∼ 1 1 (or analytic) then either |A/G| ≤ א0 or there is a perfect set of orbit inequivalent points in A. Thus we have TVC(G,Σ∼ 1 1) for each of the group in the class mentioned in 0.1-0.4 above. On the other hand, and in contrast to the usual topological Vaught conjecture, that merely asserts that 0.1-0.4 hold for arbitrary Polish groups, it is known that TVC(S∞,Σ∼ 1 1) fails. Here it is shown that the presence of S∞ is a necessary condition for TVC(G,Σ∼ 1 1) to fail: 0.5 Theorem If G is a Polish group on which the Vaught conjecture fails on analytic sets then there is a closed subgroup of G that has S∞ as a continuous homomorphic image.