Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Quotient Spaces and Homeomorphisms

  1. Feb 27, 2013 #1
    If I have the unit sphere and I mod out its equator, I get two spheres touching at one point. I have been thinking what the bijection between these could be but can not come up with one.
  2. jcsd
  3. Feb 27, 2013 #2


    User Avatar
    Science Advisor
    Gold Member

    I assume you mean S2?

    Then use the standard embedding as:

    {(x,y,z): x2+y2+z2=1}

    Send (x,y,z) to (x,y,0) to get an injection into the x-y axis. Do this for both the upper-
    and lower hemisphere, and for the equator . Notice that the set {(x,y,0)}
    in S2 will be fixed points.
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Similar Threads - Quotient Spaces Homeomorphisms Date
Quotient space of the unit sphere Nov 22, 2014
Can a nontrivial quotient space of R be homeomorphic to R? Mar 19, 2013
Quotient space of a triangle? Jan 16, 2013
Quotient Topology and Adjunction Space Sep 28, 2012
Quotient space Jun 19, 2012