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.

This implies that the Lusternik–Schnirelmann category of ⁠⁠ is at least ⁠⁠ (and it is not difficult to show that ⁠⁠ sets suffice, so the category is exactly ⁠⁠).

It is named after Lazar Lyusternik and Lev Schnirelmann, who published it in 1930.

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