Definably complete and Baire structures and Pfaffian closure - Archive ouverte HAL Accéder directement au contenu
Article Dans Une Revue Fundamenta Mathematicae Année : 2010

Definably complete and Baire structures and Pfaffian closure

Résumé

We consider definably complete and Baire expansions of ordered fields: every definable subset of the domain of the structure has a supremum and the domain can not be written as the union of a definable increasing family of nowhere dense sets. Every expansion of the real field is definably complete and Baire. So is every o-minimal expansion of a field. However, unlike the o-minimal case, the structures considered form an elementary class. In this context we prove a version of Kuratowski-Ulam's Theorem and some restricted version of Sard's Lemma. We use the above results to prove the following version of Wilkie's Theorem of the Complement: given a definably complete Baire expansion K of an ordered field with a family of smooth functions, if there are uniform bounds on the number of definably connected components of quantifier free definable sets, then K is o-minimal. We further generalize the above result, along the line of Speissegger's theorem, and prove the o-minimality of the relative Pfaffian closure of an o-minimal structure inside a definably complete Baire structure.

Dates et versions

hal-03263136 , version 1 (17-06-2021)

Identifiants

Citer

Antongiulio Fornasiero, Tamara Servi. Definably complete and Baire structures and Pfaffian closure. Fundamenta Mathematicae, 2010, 209 (3), pp.215-241. ⟨10.4064/fm209-3-2⟩. ⟨hal-03263136⟩
9 Consultations
0 Téléchargements

Altmetric

Partager

Gmail Facebook X LinkedIn More