Paper 2022 Q2

(a) Define what it means for a subset $Y$ of $\mathbb{R}^n$ to be convex. Given a topological space $X$, show that any two maps $f, g: X \rightarrow Y$ are homotopic. Define when a space is contractible, and prove that every convex subset $Y$ of $\mathbb{R}^n$ is contractible. (b) Show that $S^2$ is not homeomorphic to $S^n$ for any $n \neq 2$. (c) Show that no open subset $U$ of $\mathbb{R}^2$ is homeomorphic to $\mathbb{R}^n$ for $n \neq 2$. [Hint: If $x \in U$, then prove that $U \backslash\{x\}$ retracts onto a circle.] [You may use without proof that $\pi_1\left(S^1\right) \cong \mathbb{Z}$ and $\pi_1\left(S^n\right)=0$ for $n>1$.]