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.