TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 352, Number 12, Pages 5435–5483 S 0002-9947(00)02633-7 Article electronically published on April 13, 2000 WEAKLY O-MINIMAL STRUCTURES AND REAL CLOSED FIELDS DUGALD MACPHERSON, DAVID MARKER, AND CHARLES STEINHORN Abstract. 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 that every weakly o-minimal ordered field is real closed. We also develop a sub- stantial theory of definable sets in weakly o-minimal structures, patterned, as much as possible, after that for o-minimal structures. 1. Introduction A linearly ordered structure M =(M,<,... ) is o-minimal if every definable 1 subset of M is the union of finitely many points and open intervals in (M,<) with endpoints in M ∪ {−∞, ∞}. O-minimality has been the subject of extensive research over the last fifteen years and is the subject of the monograph [9]. This paper concerns weakly o-minimal structures M =(M,<,... ), where the hypothesis of o-minimality is relaxed to permit definable subsets of M to be the union of finitely many convex sets in (M,<). We also say that a complete theory T is weakly o- minimal if all its models are weakly o-minimal. Weak o-minimality was introduced by Dickmann in [6], where some algebraic examples are developed and some basic facts about weakly o-minimal ordered groups and rings are obtained. The principal result proved in this paper, Theorem 5.3, establishes that every weakly o-minimal ordered field is real closed. En route to proving this theorem, and for independent interest, we develop a substantial theory of definable sets in weakly o-minimal structures. Throughout the paper we assume that all structures are densely ordered; we make this supposition for convenience and because the primary examples—to our mind—are densely ordered. We now survey the contents of this paper, beginning with an overview of some of the important examples. Throughout the paper we assume familiarity with basic results of o-minimality as contained [22] and [14] (see also [9]). Received by the editors April 24, 1998. 2000 Mathematics Subject Classification. Primary 03C60, 03C64. The second author’s research was partially supported by NSF grant DMS-9626856, and the third author’s was partially supported by NSF grants DMS-9401723 and DMS-9704869, and SERC grant GR/H57097. 1 Throughout this paper ‘definable’ means ‘definable with parameters’. c 2000 American Mathematical Society 5435 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use