Covering map from paraboloid to one-sheeted hyperboloid

Question

I need to construct an explicit covering map between the following two sets:

The paraboloid $X = \{(x,y,z) \in \mathbb{R}^3:z = x^2 + y^2\}$ as covering space

The one-sheeted hyperboloid $\hat X = \{(x,y,z) \in \mathbb{R}^3:x^2 + y^2 - z^2 = 1\}$ as base space.

Can you give some intution on how to build this mapping and why it is a covering?

Answer 1

Here is the formalization of a possible sequence of maps:

$(x,y,z) \to (x,y) \to (cos \; 2 \pi x, sin \; 2 \pi x,y) \to (\sqrt{y^2+1} cos \; 2 \pi x, \sqrt{y^2+1} sin \; 2 \pi x,y)$

This is continuous and surjective. How can I proof that it is a covering space?

The answer is because the first and the last are homeomorphisms and the one in the middle is a covering map. Therefore, the whole thing must be a covering map.