Why is it impossible to cover a sphere that has radius $R$ with $3$ open semispheres of radius $R$? In my mind I have the pictorial image of the situation, but I can't find a formal proof.
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$\begingroup$ What do you mean by cover ? $\endgroup$– AmrNov 10, 2012 at 12:19
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$\begingroup$ If $S$ is the sphere (so a suface), and $U_i$ are the semispheres (so surfaces), I mean that $S=\bigcup_{i=1}^3 U_i$ $\endgroup$– DubiousNov 10, 2012 at 12:24
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$\begingroup$ ok. Now I understand. $\endgroup$– AmrNov 10, 2012 at 12:29
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$\begingroup$ Open means an open set (topological)? $\endgroup$– AmrNov 10, 2012 at 12:31
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$\begingroup$ yes I mean open set. $\endgroup$– DubiousNov 10, 2012 at 12:45
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1 Answer
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The union of two such semi-spheres always leaves an antipodal pair uncovered (where their boundaries intersect). This pair cannot be covered by a third semi-sphere since that does not contain any antipodal pair.
