Let $\tau$ be the collection of subsets of $\mathbb{C}$ of the form $\mathbb{C} \backslash f^{-1}(0)$, where $f$ is an arbitrary complex polynomial. Show that $\tau$ is a topology on $\mathbb{C}$.

Given topological spaces $X$ and $Y$, define the product topology on $X \times Y$. Equip $\mathbb{C}^{2}$ with the topology given by the product of $(\mathbb{C}, \tau)$ with itself. Let $g$ be an arbitrary two-variable complex polynomial. Is the subset $\mathbb{C}^{2} \backslash g^{-1}(0)$ always open in this topology? Justify your answer.