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anyone can help me with this problem, how can I prove that there is an homeomorphism between the closure of the (sphere minus one point) and the sphere itself any ideas?

thank you very much

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    Do you know what the closure of $S^2\setminus\{p\}$ looks like?2012-09-27
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    It seems to me that closure(sphere minus a point)=sphere.2012-09-27
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    @Mercy: Exactly. Which makes it pretty easy to find the homeomorphism: you just have to show that the closure adds only one point, and that that point ‘looks’ like the one that you removed.2012-09-27

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HINT: Show that $\operatorname{cl}_{\Bbb R^3}(S^2\setminus\{p\})=S^2$.

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    thank you for the replies, I have the same intuitions as you all, but I don't know how to show this formally. any ideas?2012-09-27
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    Show the two sets are the same, and use the identity mapping.2012-09-27
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    @user42912: Let $X=S^2\setminus\{p\}$. Let $x\in\Bbb R^3\setminus X$, and show that if $x\ne p$, then $x\notin\operatorname{cl}_{\Bbb R^3}X$, while $p\in\operatorname{cl}_{\Bbb R^3}X$. HINT: If $x\in\Bbb R^3\setminus S^2$, then $\|x\|\ne 1$.2012-09-27
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    thank you all, I will think about it.2012-09-28