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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm209-3-2,
author = {Antongiulio Fornasiero and Tamara Servi},
title = {Definably complete Baire structures},
journal = {Fundamenta Mathematicae},
volume = {209},
year = {2010},
pages = {215-241},
zbl = {1233.03043},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm209-3-2}
}
Antongiulio Fornasiero; Tamara Servi. Definably complete Baire structures. Fundamenta Mathematicae, Tome 209 (2010) pp. 215-241. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm209-3-2/