# Quotient Spaces and Homeomorphisms

1. Feb 27, 2013

### 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.

2. Feb 27, 2013

### 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.