# Question about a definition in "Topology from the differentiable viewpoint" (Milnor)

• Sep 22nd 2011, 02:25 AM
engmaths
Question about a definition in "Topology from the differentiable viewpoint" (Milnor)
Hello everyone,

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

On page one, he defines what a smooth manifold of dimenson m is. 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 $\subset$ IR^k is called a smooth manifold of dimensoin m if each x $\in$M has a neighborhood W $\cap$M that is diffeomorphic to an open subset U of the euclidian space IR^m."

What does he mean when he writes "neighborhood W $\cap$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 $\cap$M (as a subset of M) is diffeomorphic to U and U is open, then W $\cap$M is open (in M) and therefore W is an open set in IR^k which contains x. So W would be an open neighborhood of x in IR^k.

Can someone give me an (accurate) explanation?

engmaths
• Sep 22nd 2011, 05:44 AM
xxp9
Re: Question about a definition in "Topology from the differentiable viewpoint" (Miln
You're right that W is a open neighborhood of x in R^k
• Sep 23rd 2011, 04:07 AM
engmaths
Re: Question about a definition in "Topology from the differentiable viewpoint" (Miln
I made a mistake in my first post (seems that I can't edit). If W $\cap$M is open in M, it doesn't mean that W is open in R^k.

Choose R^k = R, M=(-1,1), W=(0,2], then W $\cap$M=(0,1) and therefore open in M, but W isn't open in R.
• Sep 23rd 2011, 05:43 AM
xxp9
Re: Question about a definition in "Topology from the differentiable viewpoint" (Miln
right W is a neighborhood of x in R^k, not necessarily open, but that doesn't matter.