(i) Define the product topology on $X \times Y$ for topological spaces $X$ and $Y$, proving that your definition does define a topology.

(ii) Let $X$ be the logarithmic spiral defined in polar coordinates by $r=e^{\theta}$, where $-\infty<\theta<\infty$. Show that $X$ (with the subspace topology from $\mathbf{R}^{2}$ ) is homeomorphic to the real line.