Homeomorphism $A_n=\{ x \in \mathbb R^n \;|\; 0<|x|\leq 1 \}$ and $B_n=\{ x \in \mathbb R^n \;|\;1\leq |x| \}$

Question

Let $A_n=\{ x \in \mathbb R^n \;|\; 0<|x|\leq 1 \}$ and $B_n=\{ x \in \mathbb R^n \;|\;1\leq |x| \}$, Are $A_n, B_n $ Homeomorphic?

Answer 1

I invite you to prove that the function $f\colon B_n\to A_n$ given by $f(x) = \dfrac{2x}{1+|x|}$ is a homeomorphism.

Another (maybe more intuitive) way to see it is the following: looking at $B_n$ as a subspace of $\mathbb{R}^n$ and knowing that the one point compactification of $\mathbb{R}^n$ is $\mathbb{S}^n$ (via stereographic projection), we can see that $B_n$ is taken to a punctured disk in $\mathbb{S}^n$, which is obviously homeomorphic to $A_n$.