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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Actually, "manifold" is not required. A theorem of Karl Menger*
*Or, you may prefer to say: stated by Menger, then proved by S. Lefschetz (1931) and independently by G. Nöbeling (1930).