The following question is from Fred H. Croom's book "Principles of Topology"
Prove that each open ball $B(a,r), a\in \mathbb{R}^n, r>0$, considered as a subspace of $\mathbb{R}^n$, is homeomorphic to $\mathbb{R}^n$.
After much studying, I concluded the first way to approach this problem would be by showing that the unit open ball $B(\theta,1)$ with center and radius 1 is homeomorphic to $\mathbb{R}^n$. Afterwards, show how any open ball $B(a,r)$ is topologically equivalent to $B(\theta,1)$, thus ending my proof. However, I am having a hard time showing that $B(\theta,1)$ is homeomorphic to $\mathbb{R}^n$. Any suggestions?
I want to thank you for taking the time to read this question. I greatly appreciate any assistance you provide.
Best Answer
Hint: Write $\|x\|=\sqrt{x_1^2+\dotsm+x_n^2}$ for any $x=(x_1,\dotsc,x_n)\in\Bbb R^n$. Then define a map $$f:\Bbb R^n\to B(\theta,1)$$ by $$f(x)=\frac{x}{1+\|x\|}$$
for $x\in \Bbb R^n$. Show that $f$ is continuous and bijective and the inverse map $g:B(\theta,1)\to \Bbb R^n$ defined by $$g(x)=\frac{x}{1-\|x\|}$$ is also continuous.