1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homeomorphism math help needed

  1. Nov 8, 2009 #1
    If [itex]D^n[/itex] is the unit n ball in Euclidean n-space. i.e.
    [itex]D^n = \{ x \in \mathbb{R}^n : ||x|| \leq 1 \}[/itex]
    and [itex]S^n[/itex] is an n-sphere.

    how do i show that [itex]D^n / S^{n-1}[/itex] is homeomorphic to [itex]S^n[/itex]?
    there's a hint suggesting i first of all try the n=1,2,3 cases. where [itex]X/Y= X \backslash Y \cup \{ t \}[/itex] where [itex]t \in X[/itex] is a single distinguished point.

    i'm not really sure how to start.
    i can visualise for example the n=1 case where [itex]D^1 \backslash S^0=[x-1,x) \cup (x,x+1][/itex] and [itex]S^1[/itex] is the unit circle so if we take t=x then [itex]D^1/S^0=[x-1,x+1][/itex]and i was thinking a homeomorphism here would be some sort of function along the lines of [itex]f(y)= \left( \cos{(y+1) \pi},\sin{(y+1) \pi} \right)[/itex]. i think this works as a homeomorphism but do i have to run through the steps of showing it's bijective and continuous and that the inverse is continuous?
     
    Last edited: Nov 8, 2009
  2. jcsd
  3. Nov 8, 2009 #2
    Re: Homeomorphism

    Follow the hint. Think about for example, the The circle minis one point being homeomorphic to the line . You should be able to be explicit about this homeomorphism. It makes sense right? take away one point, and then "fold"" the surface out. Show that R is homeomorphic to any open interval (this is also easy and explicit).

    Think about this for a while.
     
  4. Nov 8, 2009 #3
    Re: Homeomorphism

    is the homeomorphism i gave above wrong and if so why? also, why am i considering the circle minus one point? surely i want the line minus one point unioned with one point to be homeomorphic to the enitre circle?
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Homeomorphism math help needed
  1. Math Help needed (Replies: 13)

Loading...