# homeomorphism between the closure of the sphere minus one point and the sphere itself

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? –  Brian M. Scott Sep 27 '12 at 23:23
It seems to me that closure(sphere minus a point)=sphere. –  Mercy Sep 27 '12 at 23:24
@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. –  Brian M. Scott Sep 27 '12 at 23:26

## 1 Answer

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? –  user42912 Sep 27 '12 at 23:34
Show the two sets are the same, and use the identity mapping. –  Stefan Smith Sep 27 '12 at 23:46
@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 $\|x\|\ne 1$. –  Brian M. Scott Sep 27 '12 at 23:54
thank you all, I will think about it. –  user42912 Sep 28 '12 at 0:52