Hello everyone,

currently I'm reading the book "Topology from the differentiable viewpoint" by John Milnor.

On page one, he defines what asmooth manifold of dimenson mis. I already know some other definitions of this, but they are (slightly) different. Also, I'm not sure if I understand his definition.

He writes:

"A subset M $\displaystyle \subset$ IR^k is called a smooth manifold of dimensoin m if each x$\displaystyle \in$M has a neighborhood W$\displaystyle \cap$M that is diffeomorphic to an open subset U of the euclidian space IR^m."

What does he mean when he writes "neighborhood W$\displaystyle \cap$M"?

I know what a neighborhood is, but I don't understand what he means here. Is W a neighborhood of x in IR^k?

If W$\displaystyle \cap$M (as a subset of M) is diffeomorphic to U and U is open, then W$\displaystyle \cap$M is open (in M) and therefore W is an open set in IR^k which contains x. So W would be anopenneighborhood of x in IR^k.

Can someone give me an (accurate) explanation?

Thanks in advance!

engmaths