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A collection C of finite L-structures is a 1-dimensional asymptotic class if for every m ∈ N and every formula ϕ(x, ¯ y), where ¯ y = (y 1 ,. .. , y m): (i) There is a positive constant C and a finite set E ⊂ R >0 such that for every M ∈ C and ¯ a ∈ M m , either |ϕ(M, ¯ a)| ≤ C, or for some µ ∈ E, |ϕ(M, ¯ a)| − µ|M | ≤ C|M | 1 2. (ii) For every µ ∈ E, there(More)
We introduce the notion of definable compactness and within the context of o-minimal structures prove several topological properties of definably compact spaces. In particular a definable set in an o-minimal structure is definably compact (with respect to the subspace topology) if and only if it is closed and bounded. We then apply de-finable compactness to(More)
Let M be an o-minimal expansion of a densely ordered group and H be a pairwise disjoint collection of dense subsets of M such that H is definably independent in M. We study the structure (M, (H) H∈H). Positive results include that every open set definable in (M, (H) H∈H) is definable in M, the structure induced in (M, (H) H∈H) on any H 0 ∈ H is as simple as(More)
1. Introduction. We introduce the notion of an O-minimal theory of ordered structures, such a theory being one such that the definable subsets of its models are particularly simple. The theory of real closed fields will be an example. For T an O-minimal theory we prove that over every subset A of a model there is a prime model, which is unique up to(More)