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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.

fonini
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Kevin_H
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3 Answers3

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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.

user149418
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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.

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

Tom
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