Can manifolds be uniformly approximated by varieties?

Asked Mar 04 '11 at 23:39

Active Feb 16 '17 at 17:23

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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.

edited Feb 16 '17 at 17:23

Chill2Macht

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asked Mar 04 '11 at 23:39

Smedley Higginbottom

3

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 05 '11 at 00:52
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My idea is very fuzzy. I don't know in what way or what type of approximation. – Smedley Higginbottom Mar 05 '11 at 00:59
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Do you only care about smooth manifolds or are $C^k$-manifolds also fine? – Gerben Mar 07 '11 at 19:48

Can manifolds be uniformly approximated by varieties?

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