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I mean, is there any topological space that is locally euclidean, Haudorff and second countable and can't be embedded into a finite dimensional Euclidean space. I think it's hard for me to find such spaces because manifolds are often described visually as an euclidean subspace..

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No, there is no. This is the 'embedding thm' for manifolds. –  Berci Jan 23 '13 at 14:32
    
I'm not sure, but someone says that it's a special case, when the space is 'riemannian'.. –  Lee Dae Seok Jan 23 '13 at 15:05

1 Answer 1

Actually, "manifold" is not required. A theorem of Karl Menger*

A separable metric space that has dimension $n$ (in the sense of topological dimension) may be embedded in Euclidean space of dimension at most $2n+1$.

*Or, you may prefer to say: stated by Menger, then proved by S. Lefschetz (1931) and independently by G. Nöbeling (1930).

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