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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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What do you mean by "in the way that..." ? Given a manifold what would you like an approximation to it to mean? –  Ryan Budney Mar 5 '11 at 0:52
My idea is very fuzzy. I don't know in what way or what type of approximation. –  Smedley Higginbottom Mar 5 '11 at 0:59
Do you only care about smooth manifolds or are $C^k$-manifolds also fine? –  Gerben Mar 7 '11 at 19:48

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