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I was wondering if it is true that any open subset Ω in ℝ^{n}, to which we can associate an atlas with some coordinate charts, is always a manifold of dimensionn(the same dimension of the parent space).

Or alternatively, is it possible to find a subset of ℝ^{n}that is open, but it is a manifold of dimension lower thann?

In ℝ^{3}I cannot think of any open subset that would be a curve or a surface. So it would seem that in this caseopen subsetimplies manifold of dimension 3 (provided we can find atlas and charts to cover it).

Does this hold in general?

Thanks.

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# Open subsets and manifolds

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