 Lusternik–Schnirelmann theorem

A cover of $S^d$ by $d+1$ closed sets has a member containing an antipodal pair. The equivalent version for open sets follows by shrinking finite open covers on a compact metric space; the reverse implication follows by enlarging antipodal-pair-free closed sets slightly. This is a covering formulation of the Borsuk-Ulam theorem.

The Lusternik–Schnirelmann (LS) theorem is a result in the field of topology and calculus of variations, specifically in the context of critical point theory. It has significant implications in the study of the topology of manifolds and in variational methods. The LS theorem asserts that if a manifold is compact and has a certain topological dimension, then there exists a non-empty set of critical points for any smooth function on that manifold, provided the function satisfies certain conditions.