You're right that W is a open neighborhood of x in R^k
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.
"A subset M IR^k is called a smooth manifold of dimensoin m if each x M has a neighborhood W M that is diffeomorphic to an open subset U of the euclidian space IR^m."
What does he mean when he writes "neighborhood W 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 M (as a subset of M) is diffeomorphic to U and U is open, then W 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?
Thanks in advance!
I made a mistake in my first post (seems that I can't edit). If W 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 M=(0,1) and therefore open in M, but W isn't open in R.