Structure theorem for i-minimal expansions of the real additive ordered group

Abstract

We prove that for an o-minimal expansion of the real additive group R and a set P⊂eq R of dimension 0 such that ,P is sparse, has definable choice and every definable set has interior or is nowhere dense then, for every definable set X, there is a family \Xt:\; t∈ A\ definable in R and a set S⊂eq A of dimension 0 such that X=t∈ SXt. Moreover, in the d-minimal setting, there is a finite decomposition of X into sets of the previous form such that for every t∈ S Xt is relatively open in t∈ SXt.