## Normal series

The first step towards proving the o-minimality of $\Ran$ is to show that the quantifier-free definable sets have finitely many connected components. As discussed in this post, this means (essentially) that we need to show that basic $\Pc{R}{X}$-sets have finitely many connected components. Example 1 Let $\alpha \in \NN^n$ and $r \in (0,\infty)^n$. Show that…