# Open subsets and manifolds

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.

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lavinia
Gold Member
By definition,in an open subset of Euclidean space every point has a neighborhood that is an open ball. The coordinate transformations can be taken to be the identity map.

HallsofIvy
Homework Helper
The definition of "open set" requires that, for every point, there exist a "neighborhood" of that point contained in the set. "Neighborhoods" in an n-dimensional manifold are themselves n-dimensional so the answer to your question is "no". An open set in an n-dimensional manifold must be n-dimensional.

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lavinia
Gold Member
Hello,

Or alternatively, is it possible to find a subset of ℝn that is open, but it is a manifold of dimension lower than n?

Thanks.

The concept of dimension is well defined. That is: an open subset of Euclidean space can not be open in either a higher of lower dimensional Euclidean space.

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WWGD
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