Definably complete Baire structures

Volume 209 / 2010

Antongiulio Fornasiero, Tamara Servi Fundamenta Mathematicae 209 (2010), 215-241 MSC: Primary 58A17; Secondary 03C64, 32C05, 54E52. DOI: 10.4064/fm209-3-2


We consider definably complete Baire expansions of ordered fields: every definable subset of the domain of the structure has a supremum and the domain cannot 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, and so is every o-minimal expansion of a field. Moreover, unlike the o-minimal case, the structures considered form an axiomatizable class. In this context we prove a version of the Kuratowski–Ulam Theorem, some restricted version of Sard's Lemma and a version of Khovanskii's Finiteness Theorem. We apply these results to prove the o-minimality of every definably complete Baire expansion of an ordered field with any family of definable Pfaffian functions.


  • Antongiulio FornasieroInstitut für Mathematische Logik
    Einsteinstr. 62
    48149 Münster, Germany
  • Tamara ServiCentro de Matemática
    e Aplicações Fundamentais
    Av. Prof. Gama Pinto 2
    1649-003 Lisboa, Portugal

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