Lecture Notes in Real Algebraic and Analytic Geometry

We presume throughout some familiarity with basic model theory, in particular with the notion of a definable set. An excellent reference is [24]. A dense linearly ordered structure M= (M,<, . . .) is o-minimal (short for ordered-minimal) if every definable set (with parameters) is the union of finitely many points and open intervals (a, b), where a < b and a, b ∈ M ∪ {±∞}. The “minimal” in o-minimal reflects the fact that the definable subsets in one variable of such a structure M form the smallest collection possible: they are exactly those sets that must be definable in the presence of a linear order. This definition is the ordered analogue of minimal structures, those whose definable sets are finite or cofinite, that is, whose definable sets are those that must be definable (in the presence of equality) in every structure. The more familiar strongly minimal structures have the property every elementarily equivalent structure is minimal. Not every minimal structure is strongly minimal; see 2.9 below for the surprising situation in the ordered context.