A quasianalytic class with weakly smooth germs

TL;DR

This paper revisits a previously constructed o-minimal structure that lacks smooth cell decomposition, providing a complete proof of its o-minimality and discussing its properties related to weakly smooth germs.

Contribution

It offers a complete proof of the o-minimality of a specific structure that does not admit smooth cell decomposition, clarifying previous incomplete arguments.

Findings

The structure is o-minimal but lacks smooth cell decomposition.

The proof of o-minimality is now complete and rigorous.

The structure involves weakly smooth germs.

Abstract

In [O. Le Gal, J.-P. Rolin. An o-minimal structure which does not admit $C^\infty$ cellular decomposition. In: Ann. Inst. Fourier 59 (2009), pp 543-562], the authors construct an o-minimal structure which does not admit smooth cell-decomposition. We explain in an appendix that their proof of o-minimality is incomplete and we give a complete proof in the main text.