  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.
