We offer a topological treatment of scattered theories intended to help to explain the parallelism between, on the one hand, the theorems provable using Descriptive Set Theory by analysis of the space of countable models and, on the other, those provable by studying a tree of theories in a hierarchy of fragments of infinintary logic. We state some theorems which are, we hope, a step on the road to fully understanding counterexamples to Vaught's Conjecture. This framework is in the early (...) stages of development, and one area for future exploration is the possibility of extending it to a setting in which the spaces of types of a theory are uncountable. (shrink)

Let G be a closed subgroup of S∞ and X be a Polish G -space. To every x ∈ X we associate an admissible set Ax and show how questions about X which involve Baire category can be formalized in Ax.

Generalizing model companions from model theory we define companions of pieces of canonical partitions of Polish G-spaces. This unifies several constructions from logic. The central problem of the paper is the existence of companions which form a G-orbit which is a Gδ-set. We describe companions of some typical G-spaces.