On atomic or saturated sets

Assume T is stable, small and Phi(x) is a formula of L(T). We study the impact on T inverted right perpendicular Phi of naming finitely many elements of a model of T. We consider the cases of T inverted right perpendicular Phi which is omega-stable or superstable of finite rank. In these cases we prove that if T has < 2(aleph 0) countable models and Q = Phi(M) is countable and atomic or saturated, then any good type in S(Q) is tau-stable. If T inverted right perpendicular Phi is omega-stable and (bounded, 1-based or of finite rank) with I(T, aleph(0)) < 2(aleph 0), then we prove that every good p is an element of S(Q) is tau-stable for any countable Q. The proofs of these results lead to several new properties of small stable theories, particularly of types of finite weight in such theories.