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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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    What do you mean by cover ?2012-11-10
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    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$2012-11-10
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    ok. Now I understand.2012-11-10
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    Open means an open set (topological)?2012-11-10
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    yes I mean open set.2012-11-10

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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.