Expansions of O-minimal Structures by Dense Independent Sets, preprint 2012

Abstract. Let M be an o-minimal expansion of a densely ordered group and H be a pairwise disjoint collection of dense subsets of M such that ⋃H is definably independent in M. We study the structure (M, (H)H∈H). Positive results include that every open set definable in (M, (H)H∈H) is definable in M, the structure induced in (M, (H)H∈H) on any H0 ∈ H is as simple as possible (in a sense that is made precise), and the theory of (M, (H)H∈H) eliminates imaginaries and is strongly dependent and axiomatized over the theory of M in the most obvious way. Negative results include that (M, (H)H∈H) does not have definable Skolem functions and is neither atomic nor satisfies the exchange property