Abstract.Certain basic concepts of geometrical stability theory are generalized to a class of closure operators containing algebraic closure. A specific case of a generalized closure operator is developed which is relevant to Vaught's conjecture. As an application of the methods, we proveTheorem A.Let G be a superstate group of U-rankωsuch that the generics of G are locally modular andTh(G)has few countable models. Let G−be the group of nongeneric elements of G.G+=Go+G−.LetΠ = {q∈S(∅):U(q)<ω}.For any countable model M ofTh(G)there is a finite (...) A⊂M such thai M is almost atomic over A∪ (G+∩M) ∪ ⋃p∈Πp(M). (shrink)

It follows directly from Shelah’s structure theory that if T is a classifiable theory, then the isomorphism type of any model of T is determined by the theory of that model in the language L∞,ω1. Leo Harrington asked if one could improve this to the logic L∞, In [S. Shelah, Characterizing an -saturated model of superstable NDOP theories by its L∞,-theory, Israel Journal of Mathematics 140 61–111] Shelah gives a partial positive answer, showing that for T a countable superstable NDOP (...) theory, two -saturated models of T are isomorphic if and only if they have the same L∞,-theory. We give here a negative answer to the general question by constructing two classifiable theories, each with 21 pairwise non-isomorphic models of cardinality 1, which are all L∞,-equivalent, a shallow depth 3 ω-stable theory and a shallow NOTOP depth 1 superstable theory. In the other direction, we show that in the case of an ω-stable depth 2 theory, the L∞,-theory is enough to describe the isomorphism type of all models. (shrink)