The Torus is compact $T= -1\leq z\leq 1,x^{2}+y^{2}=\left ( 2-\lambda \right )^{2}$ where $\lambda ^{2}= 1-z^{2}$

Question

Sorry guys, but i'm stuck on this problem since a week ago, trying to figure out how to proof this Torus is compact by Heine-Borel Theorem, i guess the hardest part of it, is to proof is closed, i tried to make it by succesions, thinking that the closure is equal to the Torus itself, but by the way it's defined i guess it pretty complicated to do that. I been searching on Topology and Analysis books but there's no much useful things, also i can't see the Torus as cartesian Product. $T = (x, y, z)~\in~\mathbb{R}^{3} -1\leq z\leq 1,~x^{2}+y^{2}=\left ( 2-\lambda \right )^{2}$ where $\lambda^{2}=1-z^{2}$ Anyone can give an advice or teach me how to do it?

Answer 1

The function $$f: \mathbb R^3 \to \mathbb R^2, \quad (x, y, z) \mapsto \left(x^2 + y^2 - \left(2-\sqrt{1-\min\{1,|z|\}^2}\right)^2, z\right)$$

is continuous and as $A := \{0\} \times [-1,1]$ is closed, so is $T = f^{-1}(A)$.