Quotient space of the unit sphere

[hedipaldi]

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.

 

[gopher_p]

(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.

 

[mathwonk]

nice answer.

 

[hedipaldi]

(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