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A subset of the sphere is said short if it is contained in an openhemisphere. A short closed set which is geodesically convex is called a cap.The following theorem holds: 1. The minimal number of short closed setscovering the $n$-sphere is $n+2$. 2. If $n+2$ short closed sets cover the$n$-sphere then (i) their intersection is empty; (ii) the intersection of anyproper subfamily of them is non-empty. In the case of caps (i) and (ii) arealso sufficient for the family to be a covering of the sphere.

Sphere covering by minimal number of caps and short closed sets