# cover a sphere with 3 open semispheres

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.

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