Weakly o-minimal structures and real closed fields

Macpherson, Dugald; Marker, David; Steinhorn, Charles

doi:10.1090/S0002-9947-00-02633-7

Weakly o-minimal structures and real closed fields
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by Dugald Macpherson, David Marker and Charles Steinhorn PDF

Trans. Amer. Math. Soc. 352 (2000), 5435-5483 Request permission

Abstract:

A linearly ordered structure is weakly o-minimal if all of its definable 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 substantial theory of definable sets in weakly o-minimal structures, patterned, as much as possible, after that for o-minimal structures.

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