## Two consequences for expansions of the real field

To wrap up the notes on o-minimal structures, let’s consider two questions that were still open at the end of the last millennium: Questions Does every o-minimal expansion of the real field admit analytic cell decomposition? Is there a unique maximal o-minimal expansion of the real field? The answer to both questions is “no”, and…

## Second theorem of the complement

Let $\Delta_n$ be a collection of subsets of $\RR^n$, for $n \in \NN$, and set $\Delta = (\Delta_n)_{n \in \NN}$. As usual, we refer to the elements of the various $\Delta_n$ as $\Delta$-sets, and we call $\Delta$-sets that are manifolds $\Delta$-manifolds. We assume the following axioms for $\Delta$-sets: $(\Delta 1)$ every semialgebraic set is a…

## Global sub-$\C$-sets

Finally, we get to Step 3 of the proof of this theorem: establishing a theorem of the complement for the “right” collection of existentially definable sets. We start with a few exercises. Exercises Let $\tau_n:\RR^n \into (-1,1)^n$ be the semialgebraic diffeomorphism defined here. Show that, for every semialgebraic set $A \subseteq \RR^n$, the set $\tau_n(A)$…

## Fiber cutting

We now want to obtain a description of the projections of bounded semi-$\C$-sets. By this corollary, we only need consider projections of trivial semi-$\C$-sets, that is, of trivial $\C$-sets. We start with a few remarks about $\C$-manifolds: let $M \subseteq \RR^n$ be a nonempty $\C$-manifold of dimension $m$. First, let $r$ be a polyradius associated…

## Restricted $\C$-functions and o-minimality

We continue working with our rings of germs $\C_n$, for $n \in \NN$, as introduced here. As with convergent power series, for $f \in \C_n$ such that the polyradius $(1, \dots 1)$ is $f$-admissible, we define a function $\bar f:\RR^n \into \RR$ by setting \bar f(x):= \begin{cases} f(x) &\text{if } x \in \bar B(1), \\…