Which of the following subspaces of Euclidean space are connected? Justify your answers (i) $\left\{(x, y, z) \in \mathbf{R}^{3}: z^{2}-x^{2}-y^{2}=1\right\}$; (ii) $\left\{(x, y) \in \mathbf{R}^{2}: x^{2}=y^{2}\right\}$; (iii) $\left\{(x, y, z) \in \mathbf{R}^{3}: z=x^{2}+y^{2}\right\}$.