Charles Steinhorn

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The open core of an expansion of a dense linear order is its reduct, in the sense of definability, generated by the collection of all of its open definable sets. In this paper, expansions of dense linear orders that have o-minimal open core are investigated, with emphasis on expansions of densely ordered groups. The first main result establishes conditions(More)
A collection C of finite L-structures is a 1-dimensional asymptotic class if for every m ∈ N and every formula φ(x, ȳ), where ȳ = (y1, . . . , ym): (i) There is a positive constant C and a finite set E ⊂ R>0 such that for every M ∈ C and ā ∈ Mm, either |φ(M, ā)| ≤ C, or for some μ ∈ E, ∣∣|φ(M, ā)| − μ|M |∣∣ ≤ C|M | 2 . (ii) For every μ ∈ E, there is an(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 ominimal structure is definably compact (with respect to the subspace topology) if and only if it is closed and bounded. We then apply definable compactness to(More)