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 $G \colon X \to \mathbb {R}^m$, it will be shown that there exists an open dense semi-algebraic subset $\mathscr {U}$ 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 $F \in \mathscr {U}$, 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 $C_\infty X \cap \mathbb {S}^{n - 1}$.