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Let $D$ be the unit disk, and $E$ be its exterior, that is, $$ D=\{(x,y)\in\mathbb{R}^{2}:\quad x^2+y^2<1\}, $$ $$ E=\{(x,y)\in\mathbb{R}^{2}:\quad x^2+y^2>1\}. $$

I have two questions.

i) Is $E$ homeomorphic to $\mathbb{R}^{2}\setminus\{(0,0)\}$ ?

ii) Is $E$ homeomorphic to $D$ ?

I think that the answer to the second question is no. But I do not khow why $E$ is not homeomorphic to $D$.

And what about the first question on homeomorphism between $E$ and $\mathbb{R}^{2}\setminus\{(0,0)\}$ ? Does it exist?

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$E$ is not homeomorphic to $D$, because $D$ is simply connected, while $E$ is not.

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For (i) Define $\phi: (0,\infty) \to (1,\infty)$ by $\phi(x) = 1+x$.

Then define $\eta : \mathbb{R}^{2}\setminus\{(0,0)\} \to E$ by $\eta(x) = \frac{\phi(\|x\|)}{\|x\|} x$ is a suitable homeomorphism.

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