## Pfaffian closure

Let $\R$ be an o-minimal expansion of the real field, and let $\R_1$ be the expansion of $\R$ by all Rolle leaves over $\R$. Theorem The expansion $\R_1$ of $\R$ is o-minimal. Proof. Let $\Lambda$ be the system of all Rolle sets over $\R$; by this post, $\Lambda$ satisfies Axioms 1–7 of this post. $\qed$…

## A first set of axioms for o-minimality

Let $\Lambda = (\Lambda_n)_{n \in \NN}$ be a system of collections $\Lambda_n$ of subsets of $\RR^n$. A set $A \subseteq \RR^n$ is a $\Lambda$-set if $A \in \Lambda_n$. We let $\RR(\Lambda)$ be the expansion of the real field by all $\Lambda$-sets. We assume the following axioms for $\Lambda$: $(\Lambda 1)$ all semialgebraic sets are $\Lambda$-sets;…

## Rolle sets

Let $\R$ be an o-minimal expansion of the real field. Definition A Rolle leaf over $\R$ is a Rolle leaf of a definable $(n-1)$-distribution on $\RR^n$, for some $n \in \NN$. Example The graph of $\exp$ is a Rolle leaf over $\bar\RR$. Exercise Let $C \subseteq M^n$ be a definable $C^1$-cell of dimension $m$, let…