Template:Analogous fixed-point theorems

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 can be reduced to the other result in its column.

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