I found a theorem that the $S^2$ is connected however I cannot find a proof via Google. Is there any hint how to proof that the $S^2$ sphere is connected?
Connectedness of the $S^2$ sphere
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general-topology
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2Dear uupsklick, There are lot's of ways to prove this. You should probably provide a little more information about your background in topology (e.g. what texts you have studied from) so that people know at what level to aim their hints. Regards, – 2011-07-25
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1I just started viewing topological spaces. The definition I'm trying to understand is that there are no non-intersecting, non-empty open sets $A$ and $B$ such that $M = A \cup B$. – 2011-07-25
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0One way, which I think will make sense if you know what "connected" means, would be to exhibit a continuous map from a connected space onto $S^2$. Can you think of such a map? – 2011-07-25
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0$R^2$ is connected and I can create a map such as the arcustangens to $S^2$. Am I on the correct way? – 2011-07-25
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1With $\arctan$? Below, Bruno gives a map that would work. Of course, you have to prove that $\mathbf R^3 \setminus \{0\}$ is path connected, but that's a good exercise too. Do you know this theorem yet, by the way? That the continuous image of a connected space is connected, I mean. Proving connectedness without using these auxiliary lemmas is usually painful: a book will usually prove directly that intervals in $\mathbf R$ are connected, and then try to make use of various [theorems](http://en.wikipedia.org/wiki/Connected_space#Theorems) to show it for other spaces. – 2011-07-25
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0What are you trying to do with $\arctan$? Are you trying to show that the open unit disk in $\mathbf{R}^2$ is connected? – 2011-07-25
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2Also: do you know what arc-connectedness is and that it implies connectedness? otherwise, all of the answers so far won't help much... This is precisely why it is good to explain your background! – 2011-07-25