As is know, Vaught's conjecture is a special case of topological Vaught's conjecture.
On the other hand, the Vaught's conjecture is true for the following theories:
1- $\omega$-stable theories (Shelah),
2- superstable theories of finite rank (Buechler),
3- O-minimal theories (Mayer),
and so on.
Question 1. What is the analogue of the topological Vaught's conjecture for the above theories?
Also the topological Vaught conjecture holds for some special cases, like continuous actions of nilpotent Polish groups on Polish spaces (Hjorth-Solecki) and so on.
Question 2. Is there a case of topological Vaught's conjecture which is know to be true and which implies the Vaught's conjecture for the above theories ($\omega$-stable, superstble, ...).
