- #1
mnb96
- 715
- 5
Hello,
I was wondering if it is true that any open subset Ω in ℝn, to which we can associate an atlas with some coordinate charts, is always a manifold of dimension n (the same dimension of the parent space).
Or alternatively, is it possible to find a subset of ℝn that is open, but it is a manifold of dimension lower than n?
In ℝ3 I cannot think of any open subset that would be a curve or a surface. So it would seem that in this case open subset implies manifold of dimension 3 (provided we can find atlas and charts to cover it).
Does this hold in general?
Thanks.
I was wondering if it is true that any open subset Ω in ℝn, to which we can associate an atlas with some coordinate charts, is always a manifold of dimension n (the same dimension of the parent space).
Or alternatively, is it possible to find a subset of ℝn that is open, but it is a manifold of dimension lower than n?
In ℝ3 I cannot think of any open subset that would be a curve or a surface. So it would seem that in this case open subset implies manifold of dimension 3 (provided we can find atlas and charts to cover it).
Does this hold in general?
Thanks.
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