on Boothby and differentiable sructures

Guest

Dear Sirs
I am confused by a sentence.
At page 68, Boothby (in 'An Introduction to defferentiable manifolds
....' ,2Ed) has wrote:

Let F:R -> R be defined by F(t)=t^3. Then F is C^\infty and a
homeomorphism,
but it is not a diffeomorphism... .

ok

but then he uses this map to build a C^\infty structure on R.
and then also there is a sentence:

A necessary and sufficient condition that an open set U of M, togheter
with a mapping \phi:U -> R^n, be a coordinate neighborhood is that \phi
be a diffeomorphism of U onto an open subset W of R^n.

please give a hint, so I can understand this seeming contardictory
sentences.

best regard
farshad

Robin Chapman · science forum Guru Wannabe Joined: 25 Mar 2005 Posts: 254

farshad_ir@yahoo.com wrote:

Google