  A subset of the sphere is said short if it is contained in an open
hemisphere. A short closed set which is geodesically convex is called a cap.
The following theorem holds: 1. The minimal number of short closed sets
covering 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 any
proper subfamily of them is non-empty. In the case of caps (i) and (ii) are
also sufficient for the family to be a covering of the sphere.
