Introduction

Throughout we assume T is a complete theory in a countable language L and we work within a monster model C=Ceq of T. In this survey paper we will sketch some ideas leading to the topological stability conjecture. Also we will show how this conjecture is related to stable model theory.

One of the central open problems in model theory is Vaught's conjecture, saying that if T has few (that is, < 2ℵ0) countable models, then T has countably many of them. Thus far, this conjecture was proved for ω-stable theories [SHM], for superstable theories of finite rank [Bu2] and in some other cases [Ne10] (see also [Ls] for more information on Vaught's conjecture).

Vaught's conjecture refers to countable models and it became a yardstick with which we measure the level of our understanding of them. In fact, for a model theorist Vaught's conjecture is interesting mainly because it leads to various structural theorems on countable models. Usually such theorems say that if T has few countable models, then these models can be described so that it becomes possible to count them. So usually (and also in this paper) we assume T has <2N0 countable models, or at least that T is small (meaning that is countable for all n).