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In the discussion to the MSE post an answer of @MatthewPancia with correction in a comment of @JasonDeVito would state:

Every compact metric space of covering dimension $n$ can be embedded isometrically into $\mathbb R^N$, with $N$ sufficiently large ($N$ grows roughly as an order of $n^2$)

  1. It sounds like basically any such metric space is actually a manifold. In a sense that it can locally be approximated by a vector space, which approximation would provide it with a natural manifold structure.
    Is it? May be not for the whole space, but for its sub-spaces of covering dimension $n$? May be not for any such space, but at some conditions on metrics?

  2. Let now narrow it down to Riemannian metric spaces on $\mathbb R^n$. What are conditions for them to be manifolds, smooth manifolds, smooth manifolds everywhere but countable set of sub-spaces of zero measure?

  3. How about Finsler metric spaces on $\mathbb R^n$ to be manifolds in this sense?

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MirOdin
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    (Re: «It sounds like basically any such metric space is actually a manifold.») No. A manifold is something locally isomorphic to $\mathbb R^n$, not just embeddable in $\mathbb R^n$. Arbitrary subsets of $\mathbb R^n$ can be very complicated. – Grigory M Dec 20 '14 at 21:26
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    Also, what Matthew Pancia wrote about existence of an isometric embedding is just plain wrong. – Moishe Kohan Dec 20 '14 at 21:41
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    To see a c/e to Matthew's claim, there's a 4-point metric space that can't embed isometrically into any $\Bbb R^n$. –  Dec 20 '14 at 21:45
  • If you settle for non-isometric embeddings, $N = 2n$ will do and is optimal. – Henno Brandsma Dec 20 '14 at 21:55
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    @peth: Your questions, while interesting, suffer from lack of clear thinking. I do not know what is your level of math education, but you may want to do some reading first, before asking further questions. (At the very least, a point-set topology textbook, like Munkres "Topology".) If you manage to get through it, you will never again use expressions "sounds like basically". As for your question, I can tell you few things about homology manifolds, but, given your level of math sophistication, it would be premature. – Moishe Kohan Dec 20 '14 at 22:27
  • @studiosus: Thank you for answering my questions, your reference to Nikolaev's works was very helpful and especially your last comment about locally compact $C^1$-smooth metric spaces with limited curvature being $C^1$-smooth manifolds. I am not mathematician and am only looking here for some general level of understanding and confirmation for usage of some simple constructions related in this case to Finsler manifolds in my work in theoretical physics. Sorry, I did not want to offend with vulgar language, I just wanted to stress that I am trying to make not rigorous but qualitative statements – MirOdin Dec 21 '14 at 02:58
  • -- from Welcome to Mathematics Stack Exchange page I learned that "Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields". Am I wrong to hope that I will be able to get help here in better understanding of some mathematical concepts and relations that I need in my work? – MirOdin Dec 21 '14 at 03:09
  • @peth: In order to get good answers (and one can get them at any level) you have to do some work on your questions. The general rule of thumb is that "the deeper your questions are, the more work you have to input yourself". In particular, you have to work on posting clear questions, which, I think, require more general reading on your part. (I stand by my suggestion of Munkres, given the kind of questions you are interested in.) This is level of work is not required by most questions on this site, since they tend to be of the type "how do I compute this integral," or what is "$1+\infty$". – Moishe Kohan Dec 21 '14 at 16:10
  • Your questions are quite different and it is a good thing. One more observation: Mathematics is both a science and a language. The latter fact is usually under-appreciated. But it also imposes some burden on you, since in order to speak this language, you also have to learn it. Otherwise, you end up in a situation which amounts to, say, asking about the meaning of an intricate passage from Hegel (a famous German philosopher from the 19th century), while having hard time with the basic German grammar. – Moishe Kohan Dec 21 '14 at 16:16
  • Lastly, once you have done some thinking on your part and have clear questions, I will be happy to answer them (as I said above, they are surprisingly interesting). – Moishe Kohan Dec 21 '14 at 16:18
  • @studiosus: That is why I always preferred Kant. – MirOdin Dec 24 '14 at 08:26

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