Stability of closedness of semi-algebraic sets under continuous semi-algebraic mappings

Abstract

Given a closed semi-algebraic set $X\subset {\mathbb{R}}^n$ and a continuous semi-algebraic mapping $G:X\to {\mathbb{R}}^m,$ it will be shown that there exists an open dense semi-algebraic subset $\mathcal{U}$ of $L({\mathbb{R}}^n,{\mathbb{R}}^m),$ the space of all linear mappings from ${\mathbb{R}}^n$ to ${\mathbb{R}}^m,$ such that for all $F\in \mathcal{U},$ the image $(F+G)(X)$ is a closed (semi-algebraic) set in ${\mathbb{R}}^m.$ To do this, we study the tangent cone at infinity $C_{\infty }X$ and the set $E_{\infty }X\subset C_{\infty }X$ of (unit) exceptional directions at infinity of $X.$ Specifically we show that the set $E_{\infty }X$ is nowhere dense in $C_{\infty }X\cap {\mathbb{S}}^{n-1}.$