The theorem of the complement for nested sub-Pfaffian sets

Abstract

Let R be an o-minimal expansion of the real field, and let Lnest(R) be the language consisting of all nested Rolle leaves over R. We call a set nested sub-Pfaffian over R if it is the projection of a positive Boolean combination of definable sets and nested Rolle leaves over R. Assuming that R admits analytic cell decomposition, we prove that the complement of a nested sub-Pfaffian set over R is again a nested sub-Pfaffian set over R. As a corollary, we obtain that if R admits analytic cell decomposition, then the Pfaffian closure P(R) of R is obtained by adding to R all nested Rolle leaves over R, a one-stage process, and that P(R) is model complete in the language Lnest(R).