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

Abstract

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