Open subsets and manifolds

  • Thread starter mnb96
  • Start date
  • #1
713
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.
 
Last edited:

Answers and Replies

  • #2
lavinia
Science Advisor
Gold Member
3,259
642
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.
 
  • #3
HallsofIvy
Science Advisor
Homework Helper
41,847
966
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.
 
  • Like
Likes 1 person
  • #4
lavinia
Science Advisor
Gold Member
3,259
642
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.
 
Last edited:
  • Like
Likes 1 person

Related Threads on Open subsets and manifolds

  • Last Post
Replies
5
Views
2K
  • Last Post
Replies
13
Views
4K
Replies
1
Views
2K
  • Last Post
Replies
2
Views
5K
  • Last Post
Replies
22
Views
6K
  • Last Post
Replies
17
Views
5K
  • Last Post
Replies
9
Views
2K
  • Last Post
Replies
16
Views
5K
  • Last Post
Replies
2
Views
2K
  • Last Post
Replies
5
Views
4K
Top