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Preprint Number 1665

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1665. Erik Walsberg
Externally definable quotients and NIP expansions of the real ordered additive group

Submission date: 23 October 2019


Let R be an NIP expansion of (ℝ, < , + ) by closed subsets of ℝ^n and continuous functions f : ℝ^m → ℝ^n. Then R is generically locally o-minimal. It follows that if X ⊆ ℝ^n is definable in R then the C^k-points of X are dense in X for any k ≥ 0. This follows from a more general theorem on NIP expansions of locally compact groups, which itself follows from a result on quotients of definable sets by equivalence relations which are externally definable and ⋀-definable. We also show that R is strongly dependent if and only if R is either o-minimal or interdefinable with (ℝ, < , + , B, αℤ) for some α > 0 and collection B of bounded subsets of ℝ^n such that (ℝ, < , + , B) is o-minimal.

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Full text arXiv 1910.10572: pdf, ps.

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