Topology of the real line, Euclidean space and metric spaces

By the end you’ll be able to

• Identify open, closed, bounded, compact and connected subsets of $\mathbb{R}$ and $\mathbb{R}^n$ and apply the Heine–Borel theorem
• Find interiors, closures, boundaries and limit points, including for sets such as $\mathbb{Q}$, $\{1/n\}$ and the Cantor set
• Use the facts that continuous images of compact and connected sets are compact and connected, and that preimages of open sets under continuous maps are open