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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


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    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.
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