Locally Euclidean and Topological Manifolds

  • Thread starter elarson89
  • Start date
  • #1
elarson89
20
0
Hello,

I am learning about manifolds but I am not understanding part of the definition. This is what I'm looking at for defining the n-manifold M.

(i) M is Hausdorff
(ii) M is locally Euclidean of dimension n, and
(iii) M has a countable basis of open sets

I have a problem with (ii). To the best of my knowledge this means there exists a neighborhood around an arbitrary point of M which is homeomorphic to a piece of [tex]\Re^n[/tex]. My confusion is that I had thought that the graph of the absolute value function was not euclidean at the origin. However it is clear to me that it is naturally homeomorphic to the real line, making this euclidean, and thus a manifold.

Please advise
 

Answers and Replies

  • #2
quasar987
Science Advisor
Homework Helper
Gold Member
4,793
21
The graph of the absolute function is indeed a topological manifold.

(But it is not smooth as an embedded submanifold of R^2 because of the cusp at x=0.)
 

Suggested for: Locally Euclidean and Topological Manifolds

Replies
2
Views
301
Replies
4
Views
174
Replies
10
Views
1K
  • Last Post
Replies
10
Views
3K
  • Last Post
Replies
7
Views
628
  • Last Post
Replies
1
Views
1K
Replies
4
Views
551
Replies
9
Views
2K
  • Last Post
Replies
7
Views
747
Replies
2
Views
540
Top