Submission date: 24 March 2008. Revised and expanded version: 24 April 2009.

Abstract:

(New abstract) We consider definably complete 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, 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 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.

Mathematics Subject Classification: 58A17 (Primary), 03C64, 32C05, 54E52 (Secondary)

Keywords and phrases: Pfaffian functions; Pfaffian closure; definably complete structures; Baire spaces; o-minimality.

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