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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-Nobeling-Pontryagin 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. A more direct (and I think much easier) proof is in this paper (based again on approximation by polyhedra). In wondering about other possible approaches, I don't see how Kirby-Siebenmann could help: they show that $M\times M\times\Bbb R$ is still not a PL manifold, if $n>4$; I don't know if it could be triangulable.

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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). Here is an example: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-Nobeling-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. A more direct (and I think much easier) proof is in this paper (based again on approximation by polyhedra). In wondering about other possible approaches, I don't see how Kirby-Siebenmann could help: they show that $M\times R$ is still not a PL manifold, if $n>4$; I don't know if it could be triangulable.

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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). Here is an example:

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-Nobeling-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. A more direct (and I think much easier) proof is in this paper (based again on approximation by polyhedra). In wondering about other possible approaches, I don't see how Kirby-Siebenmann could help: they show that $M\times R$ is still not a PL manifold, if $n>4$; I don't know if it could be triangulable.