What are some examples of open subsets that are not manifolds?

[mnb96]

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.

 

[lavinia]

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]

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.

 

[lavinia]

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.

 

[WWGD]

Maybe this will help :

http://en.wikipedia.org/wiki/Invariance_of_domain