If the sphere Sn is covered by n + 1 closed sets, then one of these sets contains a pair (x, −x) of antipodal points.
It is named after Lazar Lyusternik and Lev Schnirelmann, who published it in 1930.
Each variant can be proved separately using totally different arguments, but each variant can also be reduced to the other variants in its row.
Additionally, each result in the top row can be deduced from the one below it in the same column.
[4] This topology-related article is a stub.