| course |
course_year |
question_number |
tags |
title |
year |
Analysis and Topology |
IB |
0 |
IB |
2020 |
Analysis and Topology |
|
Paper 2, Section I, $2 E$ |
2020 |
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.