Quotient Spaces and Homeomorphisms

[snesnerd]

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.

 

[WWGD]

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.