Lusternik–Schnirelmann theorem

This article is about the theorem on open covers of spheres. For the theorem on simple closed geodesics on spheres, see theorem of the three geodesics.

In mathematics, the Lusternik–Schnirelmann theorem, aka Lusternik–Schnirelmann–Borsuk theorem or LSB theorem, says as follows.

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.^[1][2][3]

Equivalent results

There are several fixed-point theorems which come in three equivalent variants: an algebraic topology variant, a combinatorial variant and a set-covering variant. 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]

Algebraic topology	Combinatorics	Set covering
Brouwer fixed-point theorem	Sperner's lemma	Knaster–Kuratowski–Mazurkiewicz lemma
Borsuk–Ulam theorem	Tucker's lemma	Lusternik–Schnirelmann theorem