Results 1 to 2 of 2

Math Help - Homeomorphism with surfaces

  1. #1
    Newbie
    Joined
    Oct 2012
    From
    Chile
    Posts
    4

    Homeomorphism with surfaces

    Let H=\{(x,y,z)\in\mathbb R^3|x^2+y^2=1+z^2\}, and C=\{(x,y,z)\in\mathbb R^3|x^2+y^2=1\}. Prove that H and C as subspaces of \mathbb R^3 are homeomorph.

    How to solve this analytically? I've seen a geometric solution but I don't see how to work it analytically.
    Thanks.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member
    Joined
    Sep 2012
    From
    Washington DC USA
    Posts
    525
    Thanks
    147

    Re: Homeomorphism with surfaces

    If you understand it geometrically, then do you see what the simplest point-to-point mapping is that will give you your homeomorphism?

    What point of C should the point (3, 0, 2sqrt(2)) in H map to?

    What point in H should the point (-sqrt(3)/2, 1/2, 10) in C map to? What about the point ( 0, -1, 10) in C?

    The answer to your question is that it is possible to explicitly/analytically write down the homeomorphism, and its inverse.

    As always, do a few concrete examples, like with the points I suggested above, to help you "see" what's happening.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. homeomorphism between R^(n^2) and M_n(R)
    Posted in the Differential Geometry Forum
    Replies: 3
    Last Post: January 21st 2011, 06:20 PM
  2. homeomorphism
    Posted in the Differential Geometry Forum
    Replies: 0
    Last Post: November 30th 2009, 02:29 PM
  3. Homeomorphism
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: November 2nd 2009, 04:11 AM
  4. Homeomorphism
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: November 1st 2009, 12:20 AM
  5. homeomorphism
    Posted in the Differential Geometry Forum
    Replies: 9
    Last Post: August 12th 2009, 02:51 PM

Search Tags


/mathhelpforum @mathhelpforum