2
$\begingroup$

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.

$\endgroup$
4
$\begingroup$

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.

$\endgroup$
2
$\begingroup$

HINT: Your approach is a good one. You need a way to map $[0,1)$ nicely onto $[0,\to)$ or the reverse; how about

$$f:[0,\to)\to[0,1):x\mapsto\frac{x}{1+x}\;?$$

Try using that to convert big radial distances into small ones.

$\endgroup$
1
$\begingroup$

Hing: Take $B(0,1)$. The let $f(x) = \frac{x}{1-|x|}$.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.