Hausdorff limits of Rolle leaves

Abstract

Let \({\mathcal{R}}\) be an o-minimal expansion of the real field. We introduce a class of Hausdorff limits, the T ∞-limits over \({\mathcal{R}}\), that do not in general fall under the scope of Marker and Steinhorn’s definability-of-types theorem. We prove that if \({\mathcal{R}}\) admits analytic cell decomposition, then every T ∞-limit over \({\mathcal{R}}\) is definable in the pfaffian closure of \({\mathcal{R}}\).