Relative Pfaffian closure for Definably Complete Baire Structures

Speissegger proved that the Pfaffian closure of an o- minimal expansion of the real field is o-minimal. Here we give a first order version of this result: having introduced the notion of definably complete Baire structure, we define the relative Pfaf- fian closure of an o-minimal structure inside a definably complete Baire structure, and we prove its o-minimality. We derive effec- tive bounds on some topological invariants of sets definable in the Pfaffian closure of an o-minimal expansion of the real field.