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.
cover a sphere with 3 open semispheres
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linear-algebra
geometry
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0What do you mean by cover ? – 2012-11-10
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0If $S$ 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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0ok. Now I understand. – 2012-11-10
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0Open means an open set (topological)? – 2012-11-10
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0yes I mean open set. – 2012-11-10
1 Answers
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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.