LetTbe a countable, small simple theory. In this paper, we prove that for suchT, the notion of Lascar strong type coincides with the notion of strong type, over an arbitrary set.

Let T be a countable, small simple theory. In this paper, we prove that for such T, the notion of Lascar strong type coincides with the notion of strong type, over an arbitrary set.

We continue work of Shelah on the cardinality of families of pairwise incompatible types in simple theories obtaining characterizations of simple and supersimple theories. We develop a local analysis of the number of types in simple theories and we find a new example of a simple unstable theory.

In this paper a class of simple theories, called the low theories is developed, and the following is proved. Theorem. Let T be a low theory. A set and a, b elements realizing the same strong type over A. Then, a and b realized the same Lascar strong type over A.