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.



A senseful meaning of 'approximation of manifolds'?

> 
> A senseful meaning of ‘approximation of manifolds’?
> 
> 
> 


A simply-looking meaning is the one of Gromov-Hausdorff, as mentioned by Ryan Budney.
It has many cool applications, but none that I'm aware of are to topological manifolds.


The kind of approximation that one normally uses to prove something about topological manifolds is representing your manifold as an inverse limit of polyhedra (rather than smooth manifolds). Examples of such use are incidentally given below.


I don't think that



> 
> by Whitney's theorem, any $n$-dimensional topological manifold $M$ can be continuously embedded into the larger-dimensional euclidian space $\Bbb R^{2n}$.
> 
> 
> 


Whitney's embedding theorem applies only to smooth manifolds. To embed a non-triangulable topological manifold in $\Bbb R^{2n+1}$ (note the dimension shift) one "approximates" the manifold by polyhedra and embeds those first (using PL general position); no simpler way is known. This is called the Menger-Nöbeling-Pontryagin embedding theorem (or after various subsets of the 3 authors); a clear proof appears in *J. R. Isbell, Embeddings of inverse limits, Ann. of Math. 70 (1959), 73-84* and in Isbell's book "Uniform spaces". (Many other books give a less explicit proof based on the Baire category theorem.)


If you really want to embed a non-triangulable topological $n$-manifold $M$ in $\Bbb R^{2n}$ (and not just in $\Bbb R^{2n+1}$), this is harder. Earliest results that seem to imply this are in Bryant-Mio and Johnston's 1999 papers in Topology, available from [Ranicki's website](http://www.maths.ed.ac.uk/%7Eaar/homology). A more direct (and I think much easier) proof is [in this paper](https://arxiv.org/abs/math/0612085) (based again on approximation by polyhedra).
In wondering about other possible approaches, I don't see how Kirby-Siebenmann could help: they show that $M\times\Bbb R$ is still not a PL manifold, if $n>4$; I don't know if it could be triangulable.


