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2 clarification

Any continuous function can be uniformly approximated by smooth functions.

I would like to have something similar - in what-ever sense - for continuous manifolds.

For example, by Whitney's theorem, any $n$-dimensional topological manifold $M$ can be continuously embedded into the larger-dimensional euclidian space $\mathbb R^{2n}$. You can construct a continuous function $f$ with image $[-1,1]$ on $\mathbb R^{2n}$, whose zero level set is exactly (the image of) $M$.

A meaning of "approximating a manifold" would be to approximate such a level set function by smooth functions. However, Whitney's theorem is non constructive, you need a metric on the manifold for the question to make sense, and there are likely to appear difficulties.

Do you where to find a elaboration on questions like the above? (Of course, different approaches are of interest as well.). Thank you.

1

A senseful meaning of 'approximation of manifolds'?

Any continuous function can be uniformly approximated by smooth functions.

I would like to have something similar - in what-ever sense - for continuous manifolds.

For example, by Whitney's theorem, any $n$-dimensional topological manifold $M$ can be continuously embedded into the larger-dimensional euclidian space $\mathbb R^{2n}$. You can construct a continuous function $f$ with image $[-1,1]$ on $\mathbb R^{2n}$, whose zero level set is exactly (the image of) $M$.

A meaning of "approximating a manifold" would be to approximate such a level set function by smooth functions. However, Whitney's theorem is non constructive, and there are likely to appear difficulties.

Do you where to find a elaboration on questions like the above? (Of course, different approaches are of interest as well.). Thank you.