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It is proved that any O-minimal structure M (in which the underlying order is dense) is strongly O-minimal (namely, every N elementarily equivalent to M is O-minimal). It is simultaneously proved that if M is 0-minimal, then every definable set of n-tuples of M has finitely many "definably connected components." 0. Introduction. In this paper we study the… (More)

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)

- Ya’acov Peterzil, Charles Steinhorn
- 1999

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)

A linearly ordered structure is weakly o-minimal if all of its de-finable sets in one variable are the union of finitely many convex sets in the structure. Weakly o-minimal structures were introduced by Dickmann, and they arise in several contexts. We here prove several fundamental results about weakly o-minimal structures. Foremost among these, we show… (More)

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)

- David Marker, Charles Steinhorn
- J. Symb. Log.
- 1994

- Michael C. Laskowski, Charles Steinhorn
- J. Symb. Log.
- 1995

- Alfred Dolich, Chris Miller, Charles Steinhorn
- Ann. Pure Appl. Logic
- 2016

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)

- Dugald Macpherson, Charles Steinhorn
- Ann. Pure Appl. Logic
- 1996

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)