I need to construct a homeomorphism between $\mathbb{R}^2\sqcup\mathbb{R}^2$ and $\mathbb{S}=\{(x,y,z)\in \mathbb{R}^2| x^2-y^2-z^2=1\}$.
The general idea I have is to define
$g:\mathbb{R}^2\rightarrow \mathbb{S}$ to have the rule of assignment $g:(y,z)\longmapsto (\sqrt{y^2+z^2+1},y,z)$
and define
$h:\mathbb{R}^2\rightarrow \mathbb{S}$ to have the rule of assignment $h:(y,z)\longmapsto (-\sqrt{y^2+z^2+1},y,z)$.
But how do I denote the restrictions of my new continuous function $f:\mathbb{R}^2\sqcup \mathbb{R}^2\rightarrow \mathbb{S}$ formed on the basis of the Universal Property of Disjoint Unions?
And have I even defined the proper functions $g$ and $h$ to get a homeomorphism?
Thank you. Let me know if you need any clarifications.
Edit: I realized that I said something incorrect and deleted that part.
Your solution is OK. The space $\Bbb S$ consists of two open disjoint parts and each of the maps $f$ and $g$ provides a homeomorphism of a copy of the plane $\Bbb R^2$ with the respective part. These maps are homeomorphisms, because they are continuous are their inverse maps are projections onto two last coordinates, so continuous too. You can describe the constructed homeomorphism $h$ formally by putting its domain $\{-1,1\}\times \Bbb R^2$ and $h(x,y,z)=(x\sqrt{y^2+z^2+1},y,z)$ for each $(x,y,z)$.