Weakly o-minimal structures and real closed fields

A linearly ordered structure is weakly o-minimal if all of its definable se ts in one variable are the union of finitely many convex sets in the struct ure. Weakly o-minimal structures were introduced by Dickmann, and they aris e in several contexts. We here prove several fundamental results about weak ly 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 pos sible, after that for o-minimal structures.