Locally Euclidean and Topological Manifolds

[elarson89]

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 \Re^n. 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

 

[quasar987]

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.)