Real Algebraic and Analytic Geometry
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195. Nicolas Dutertre:
Semi-algebraic neighborhoods of closed semi-algebraic sets.

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Submission: 2006, February 13.

Abstract:
Given a closed (not necessarily compact) semi-algebraic set $X$ in $\mathbb{R}^n$, we construct a nonnegative semi-algebraic $C^2$ function $f$ such that $X=f^{-1}(0)$ and such that for $\delta > 0$ sufficiently small, the inclusion $X \subset f^{-1}([0,\delta])$ is a retraction. As a corollary, we obtain several formulas for $\chi(X)$.

Mathematics Subject Classification (2000): 14P10, 14P25.

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