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Proof. This follows easily from Lemma 5.1.17 of Buechler [7], which states that the indiscernible sequence I is a Morley sequence over C ∪J for any infinite J ⊂ I. To apply this, let i 1 , . . . , i n ∈ I be distinct, and let J 1 , J 2 be infinite disjoint subsets of Stably embedded sets.A C-definable set D in U is stably embedded if, for any definable set E and r > 0, E ∩ D r is definable over C ∪ D. If instead we worked in a small model M , and C, D were from M eq , we would say that D is stably embedded if for Proof. (i) We may suppose that A st | ⌣C M st . Then, by the hypothesis, tp(A st A/M ) is the unique extension of tp(A st A/C) ∪ tp(A st /M ) to M . Since tp(A st /M st ) is definable over acl(C) in the stable structure St C , it follows from Lemma 3.12 that tp(A/M ) is definable over acl(C). In particular, tp(A/M ) is Aut(M/acl(C))-invariant.(ii) Now suppose A ′′ ≡ acl(C) A ′ and tp(A ′ /M ), tp(A ′′ /M ) are both Aut(M/acl(C))invariant extensions of tp(A/C). Then tp((A ′ ) st /M st ) and tp((A ′′ ) st /M st ) are both invariant extensions of tp(A st /acl(C)) in St C : indeed, any automorphism in Aut(St C /acl(C)) extends to an automorphism of M over acl(C) (saturation of M and stable embeddedness), so fixes tp(A ′ /M ) and tp(A ′′ /M ), and hence fixes tp((A ′ ) st /M st ) and tp((A ′′ ) st /M st ). Hence tp((A ′ ) st /M st ) and tp((A ′′ ) st /M st ) are equal, as invariant extensions of a type over an algebraically closed base are

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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.

LEMMA 2.1. / / r /^Q and \Y 1 n F 2 | = minflFJ, |F 2 |} then U F 2 ) = Since the combinatorial argument given in [8] for the case where T x and F 2 are countably infinite needs only trivial modification to prove this we omit details. LEMMA 2.2. If all moieties ofQ are full for the subgroup G of S then G = S.This is [19, Note 3(iii) of §4] and we will not rehearse the proof here. LEMMA 2.3. Let H be a subgroup of S and let EpEg be subsets ofQ. that are full for H. If |Z X n I 2 | = K and Z x U S 2 = ft then H = S.Proof. Let I be any moiety and let Z o be a fixed moiety of E x n Z 2 . We are assuming that E x U Z 2 = ft and so £ must meet at least one of I x and I 2 in a 3 JLM42