Semi-algebraic neighborhoods of closed semi-algebraic sets

Étant donné un ensemble semi-algébrique fermé (non nécessairement compact) $X$ de ${ℝ}^n$, nous construisons une fonction semi-algébrique $f$ positive et de classe ${𝒞}^2$ telle que $X=f^{-1}\left(0\right)$ et telle que pour $\delta >0$ suffisamment petit, l’inclusion de $X$ dans $f^{-1}\left([0,\delta ]\right)$ soit une rétraction. En corollaire, nous obtenons plusieurs formules pour la caractéristique d’Euler de $X$.

Given a closed (not necessarly compact) semi-algebraic set $X$ in ${ℝ}^n$, we construct a non-negative semi-algebraic ${𝒞}^2$ function $f$ such that $X=f^{-1}\left(0\right)$ and such that for $\delta >0$ sufficiently small, the inclusion of $X$ in $f^{-1}\left([0,\delta ]\right)$ is a retraction. As a corollary, we obtain several formulas for the Euler characteristic of $X$.

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