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.