Externally definable quotients and NIP expansions of the real ordered additive group

Let R\mathscr {R} be an NIP\mathrm {NIP} expansion of (R,>,+)(\mathbb {R},>,+) by closed subsets of Rn\mathbb {R}^n and continuous functions f:Rm→Rnf : \mathbb {R}^m \to \mathbb {R}^n. Then R\mathscr {R} is generically locally o-minimal. This follows from a more general theorem on NIP\mathrm {NIP} expansions of locally compact groups, which itself follows from a result on quotients of definable sets in ℵ1\aleph _1-saturated NIP\mathrm {NIP} structures by equivalence relations which are both externally definable and ⋀\bigwedge-definable. We also show that R\mathscr {R} is strongly dependent if and only if R\mathscr {R} is either o-minimal or (R,>,+,αZ)(\mathbb {R},>,+,\alpha \mathbb {Z})-minimal for some α>0\alpha > 0.