Let $C = \left\{x,y,z\in \mathbb{R}\mid x^{2} + y^{2} = 1\right\}$ be a infinite cone, $H = \left\{x,y,z\in \mathbb{R}\mid x^{2} + y^{2} - z^{2}= 1\right\}$ one sheeted hyperboloid. Using cylindrical coordinates \begin{gather} C = \left\{\left(\sqrt{1+z^2}\cos(\varphi),\sqrt{1+z^2}\sin(\varphi),z\right)\mid z\in\mathbb{R},\varphi\in[0,2\pi]\right\},\\ H = \left\{\left(\cos(\varphi),\sin(\varphi),z\right)\mid z\in\mathbb{R},\varphi\in[0,2\pi]\right\}. \end{gather} So i suspect that the mapping \begin{equation} \eta: \left(\sqrt{1+z^2}\cos(\varphi),\sqrt{1+z^2}\sin(\varphi),z\right) \to \left(\cos(\varphi),\sin(\varphi),z\right) \end{equation} is a homeomorphism (moreover a diffeomorphism?) of C and H, however I don’t quite understand how to show that it is a homeomorphysm by definition. Is it enough to show that $\eta$ and $\eta^{-1}$ are continuous componentwise? (The fact that $\eta$ is one to one is obvious).
Does the homeomorphism of $C$ and $H$ entails that $H$ is connected and not compact?