Categories of Topological Spaces and Scattered Theories

Abstract

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.