Submission date: 30 April 2020

Abstract:

We prove that for an o-minimal expansion of the real additive group R and a set P ⊆ ℝ of dimension 0 such that 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 {X_t: t in A} definable in R and a set S ⊆ A of dimension 0 such that X = ∪_{t in S}X_t. Moreover, in the d-minimal setting, there is a finite decomposition of X into sets of the previous form such that for every t in S, X_t is relatively open in ∪_{t in S}X_t.

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Full text arXiv 2005.00063: pdf, ps.