论文标题
连续半代数映射下半代数集的闭合度的稳定性
Stability of closedness of semi-algebraic sets under continuous semi-algebraic mappings
论文作者
论文摘要
给定一个封闭的半代数集$ x \ subset \ mathbb {r}^n $和连续的半代价映射$ g \ g \ colon x \ to \ mathbb {r}^m，$，$，它将显示出存在一个开放式持续的持开放式的半含量sem-algebraic subset $ \ l（u} $} $} \ Mathbb {r}^m），$从$ \ Mathbb {r}^n $到$ \ m artbb {r}^m，$，使所有$ f \ in \ mathscr {u}中的$ f \ ymathscr {u}中的所有线性映射的空间这样做，我们研究了无限$ c_ \ infty x $的切线锥和$ e_ \ infty x \ subset c_ \ subset c_ \ Infty x $（单位）$ x的特殊指示的x $。
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 $C_\infty X \cap \mathbb{S}^{n - 1}.$
