Patrick Speissegger
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. $\qed$…
Let $\Sigma = (\Sigma_n)_{n \in \NN}$ be a system of collections $\Sigma_n$ of subsets of $\RR^n$. A set $A \subseteq \RR^n$ is a $\Sigma$-set if $A \in \Sigma_n$. We let $\RR(\Sigma)$ be the expansion of the real field by all $\Sigma$-sets. We assume the following axioms for $\Sigma$: $(\Sigma 1)$ all semialgebraic sets are $\Sigma$-sets;…
Let $\Lambda = (\Lambda_n)_{n \in \NN}$ be a system of collections $\Lambda_n$ of subsets of $\RR^n$ satisfying Axioms $(\Lambda 1)$–$(\Lambda 7)$. The goal of this post is to describe the topological closure and boundary of a bounded $\Lambda^\infty$-set. To fix notations below, let $X \subseteq \RR^n$ be a bounded, basic $\Lambda^\infty$-set. Let $W \subseteq \RR^{k+n+l}$…
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;…
In order to prove o-minimality for structures generated by Rolle sets, we need another tool from topology: Hausdorff limits (see Section 21 of Kuratowski’s book). Definition For compact sets $A,B \subseteq \RR^n$, the Hausdorff distance $d(S,T)$ is the greater of the two values $\sup\set{d(x,T):\ x \in S}$ and $\sup\set{d(x,S):\ x \in T}$. Note that $d_n(A,\emptyset)…