Introduction

• Distinguish between 2-sphere \(S^2\) and 2-torus \(S^1 \times S^1\).
• Homotopy rubber band on \(S^2\) contracts to a point.
• For continuous maps \(f, g\), they are homotpic \(f \simeq g\) if there exists a homotopy between them. Check equivalnce relation.
• Identify homeomorphically \(S^n\) without North Pole with \(\mathbb{R}^n\), via Stereographic projection.
• Identity map on \(\mathbb{R}^n\).
• Finite is limited: \(S^n\) is not the union of finitely many great circles, \(\mathbb{R}^n\) is not the union of finitely many 2 dimensional vector subspaces.