Enclosing a Compact Set in an O-minimal Expansion of $(\mathbb{R},+,\cdot ,0,1<)$

Abstract

Fix an o-minimal expansion $\mathcal{R}=(\mathbb{R},+,\cdot ,0,1<,...)$ of the real ordered field. Given $C^1$ functions $f_1,...,f_k$, $g_1,...,g_k:M\to \mathbb{R}$ on a definable cell $M$, let $h_{i,0}$ denote $f_i$ and $h_{i,1}$ denote $g_i$. Suppose that for all $\tau \in 2^{[k]}$, $H_{\tau }=(h_{1,\tau (1)},..,h_{k,\tau (k)}):M\to {\mathbb{R}}^k$ is regular and proper on $M$, and that for all $i\in [k]$, $\{f_i=0\}$ and $\{g_i=0\}$ are connected, and $\{f_i=0\}\cap \{g_i=0\}=\varnothing$. We show that then there exists a sequence $({\square }_{i,\epsilon }:i\in [k],\epsilon \in \{0,1\})\in \{\le ,\ge {\}}^{[k]\times \{0,1\}}$ such that the enclosed region $\bigcap _{i\in [k]}\{f_i{\square }_{i,0}0\}\cap \{g_i{\square }_{i,1}0\}$ is compact.