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Can manifolds be uniformly approximated by varieties in the way that continuous functions can be uniformly approximated by polynomials?

I got the idea from reading the Princeton companion to mathematics when it gave:

Theorem (Nash) Let $M$ be a manifold in $\mathbb{R}^n$. Fix any large number $R$. Then there is a polynomial $f$ whose zero set is as close to $M$ as we want, at least inside a ball of radius $R$ around the origin.

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    Do you only care about smooth manifolds or are $C^k$-manifolds also fine?2011-03-07

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