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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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  • $\begingroup$ What do you mean by cover ? $\endgroup$ – Amr Nov 10 '12 at 12:19
  • $\begingroup$ If $S$ is the sphere (so a suface), and $U_i$ are the semispheres (so surfaces), I mean that $S=\bigcup_{i=1}^3 U_i$ $\endgroup$ – Dubious Nov 10 '12 at 12:24
  • $\begingroup$ ok. Now I understand. $\endgroup$ – Amr Nov 10 '12 at 12:29
  • $\begingroup$ Open means an open set (topological)? $\endgroup$ – Amr Nov 10 '12 at 12:31
  • $\begingroup$ yes I mean open set. $\endgroup$ – Dubious Nov 10 '12 at 12:45
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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.

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