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Quotient space of the unit sphere

  1. Nov 22, 2014 #1
    prove that the quotient space obtained by identifying the points on the southern hemisphere, is homeomorphic to the whole sphere.I am trying to define a homeomorphism between the quotient space and the sphere,and i need help doing it.
    Thank's in advance.
  2. jcsd
  3. Nov 23, 2014 #2
    (1) The equator would necessarily need to be part of the "southern hemisphere" in order for the claim to be true.

    (2) Are you able to show that the northern hemisphere - minus the equator - is homeomorphic to the sphere less a single point? Don't worry so much yet about finding a formula for a function that does this; just get a basic idea/picture for how the homeomorphism might work.
  4. Nov 24, 2014 #3


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    nice answer.
  5. Nov 28, 2014 #4
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