Hausdorff limits of Rolle leaves

TL;DR

This paper introduces a new class of Hausdorff limits called T-infinity limits in o-minimal structures and proves their definability in the pfaffian closure under certain conditions.

Contribution

It defines T-infinity limits over o-minimal structures and establishes their definability in the pfaffian closure when analytic cell decomposition exists.

Findings

T-infinity limits are a new class of Hausdorff limits.

Such limits are definable in the pfaffian closure under analytic cell decomposition.

The results extend the scope of definability beyond Marker and Steinhorn's theorem.

Abstract

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