Homeomorphism of a Torus

Question

I need to show that the function:

$f(u,v)=((r \cos(u)+a)\cos(v), (r\cos(u)+a)\sin(v), r\sin(u))$, $f:U\to\mathbb{R^3}$, when $U=(0,2\pi)\times (0,2\pi)$ is an homeomorphism to the torus.

I show that $f$ is differentiable and then $f$ is continous, also I show that $f$ is injetive. I tried to calculate the inverse function...but I'm stucked.

Also...I tried to show that $f$ is an open function...but I don't know how I can start the proof.

Answer 1

Well, they're not homeomorphic, as $(0, 2\pi) \times (0, 2\pi)$ has a trivial fundamental group, while $S^1 \times S^1$ does not.