= Lusternik–Schnirelmann theorem
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= Lusternik-Schnirelmann-Borsuk theorem
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= Lusternik-Shnirelman-Borsuk theorem
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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>.
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