Expansions of the real line by open sets: o-minimality and open cores

The open core of a structure R := (R, <, . . .) is defined to be the reduct (in the sense of definability) of R generated by all of its definable open sets. If the open core of R is o-minimal, then the topological closure of any definable set has finitely many connected components. We show that if every definable subset of R is finite or uncountable, or if R defines addition and multiplication and every definable open subset of R has finitely many connected components, then the open core of R is o-minimal. An expansion R of the real line (R, <) is o-minimal if every definable subset of R is a finite union of points and open intervals (that is, has finitely many connected components). Such structures—particularly, o-minimal expansions of the field of real numbers—have many nice properties, and are of interest not only to model theorists, but to analysts and geometers as well. (See e.g. [D2], [DM] for expositions of the subject.) Conventions. Throughout, given A ⊆ R, “A-definable” means “definable (in the structure under consideration) using parameters from A”, and “definable” means “R-definable”. We use “reduct” and “expansion” in the sense of definability, that is, given structures R1 and R2 with underlying set R, we say that R1 is a reduct of R2—equivalently, R2 is an expansion of R1, or R2 expands R1—if every A ⊆ R definable in R1 is definable in R2, for every n ∈ N. (R0 denotes the one-point space {0}.) 1991 Mathematics Subject Classification: Primary 03C99. Research supported by the Fields Institute for Research in Mathematical Sciences and NSERC Grant OGP0009070. The first author was also supported in part by NSF Grant DMS-9896225.