Quotient space of the unit sphere

  • Thread starter hedipaldi
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  • #1
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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.
 

Answers and Replies

  • #2
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(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.
 
  • #3
mathwonk
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nice answer.
 
  • #4
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(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.
Thank's
 

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