Homeomorphism with surfaces

• Oct 18th 2012, 04:31 PM
Megus
Homeomorphism with surfaces
Let $H=\{(x,y,z)\in\mathbb R^3|x^2+y^2=1+z^2\},$ and $C=\{(x,y,z)\in\mathbb R^3|x^2+y^2=1\}.$ Prove that $H$ and $C$ as subspaces of $\mathbb R^3$ are homeomorph.

How to solve this analytically? I've seen a geometric solution but I don't see how to work it analytically.
Thanks.
• Oct 18th 2012, 10:46 PM
johnsomeone
Re: Homeomorphism with surfaces
If you understand it geometrically, then do you see what the simplest point-to-point mapping is that will give you your homeomorphism?

What point of C should the point (3, 0, 2sqrt(2)) in H map to?

What point in H should the point (-sqrt(3)/2, 1/2, 10) in C map to? What about the point ( 0, -1, 10) in C?

The answer to your question is that it is possible to explicitly/analytically write down the homeomorphism, and its inverse.

As always, do a few concrete examples, like with the points I suggested above, to help you "see" what's happening.