Definably complete and Baire structures and Pfaffian closure

TL;DR

This paper explores the properties of definably complete Baire structures, proving key theorems like Kuratowski-Ulam and Sard's Lemma, and establishes conditions under which such structures are o-minimal, including their Pfaffian closures.

Contribution

It introduces a framework for definably complete Baire structures, proves foundational theorems within this context, and extends Wilkie's Theorem to these structures, including their Pfaffian closures.

Findings

Proved a version of Kuratowski-Ulam's Theorem for these structures.

Established a restricted Sard's Lemma in this setting.

Showed that under certain conditions, these structures are o-minimal.

Abstract

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.