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I am reading about manifolds from the book by Millman. He says that manifolds locally looks like a open set, but there is no canonical way to make M look like a Euclidean Space, and so we can't define tangent vectors in general sense. My doubt is even in $\mathbb{R^n}$, we check differentiablity only in an open set near a point rather than the whole space. So, why are we concerned about the manifold not looking like Euclidean space? Also explain difference between M is locally an open set and M is locally an Euclidean space. This paragraph is from the book i am reading. enter image description here

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  • $\begingroup$ What do you mean by "M is locally an open set"? As for your question "why are we concerned..." my answer would be that having the entire $M$ an open subset of $R^n$ would eliminate the need for transition maps. Thus: We are not concerned. $\endgroup$ – Moishe Kohan Mar 13 '14 at 20:10

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