Prove that $A$ is homeomorphic to $S^1\times[1,2]$ where $A$ and $S^1$ are defined below:

Question

let A = $\{(x,y) \in \mathbb R^2 : 1 \le \sqrt{x^2+y^2} \le 2 \}$ . Prove that $A$ is homeomorphic to $S^1\times[1,2]$ where $S^1 = \{(x,y) \in \mathbb R^2 : x^2+y^2 =1 \}$

Answer 1

Notice that $A=\{(rcos \theta,rsin \theta): 1≤r≤2,0≤\theta<2\pi\}$. Therefore required homeomorphism is $f:A\rightarrow S^1×[1,2]$ is given by $f(rcos \theta,rsin \theta)=((cos\theta,sin\theta),r)$.To show $f$ is homeomorphism first show it is bijective and continuous. Now $A$ is compact and $S^1×[1,2]$ is hausdorff i.e. $f^{-1}$ is also continuous.