Results 1 to 4 of 4

Math Help - Question about a definition in "Topology from the differentiable viewpoint" (Milnor)

  1. #1
    Newbie
    Joined
    Sep 2011
    Posts
    22

    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 \inM has a neighborhood W \capM that is diffeomorphic to an open subset U of the euclidian space IR^m."

    What does he mean when he writes "neighborhood W \capM"?
    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 \capM (as a subset of M) is diffeomorphic to U and U is open, then W \capM 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!

    engmaths
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Senior Member
    Joined
    Mar 2010
    From
    Beijing, China
    Posts
    293
    Thanks
    23

    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
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Sep 2011
    Posts
    22

    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 \capM 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 \capM=(0,1) and therefore open in M, but W isn't open in R.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Senior Member
    Joined
    Mar 2010
    From
    Beijing, China
    Posts
    293
    Thanks
    23

    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.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 4
    Last Post: December 11th 2011, 08:40 AM
  2. Replies: 4
    Last Post: November 20th 2011, 01:57 PM
  3. Replies: 1
    Last Post: October 25th 2010, 05:45 AM
  4. Rigorous Definition of "Inequality" or "Positive" in the Reals
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: July 22nd 2010, 01:23 AM
  5. A question about definition of "Binary Relation"
    Posted in the Discrete Math Forum
    Replies: 3
    Last Post: October 16th 2009, 03:56 AM

Search Tags


/mathhelpforum @mathhelpforum