An elementary direct proof that the Knaster-Kuratowski-Mazurkiewicz lemma implies Sperner’s lemma. (English) Zbl 1398.54084
Summary: Three central results in economic theory – Brouwer’s fixed-point theorem, Sperner’s lemma, and the Knaster-Kuratowski-Mazurkiewicz (KKM) lemma – are known to be equivalent. In almost all cases, elementary direct proofs of one of these results using any of the others are easily found in the literature. This seems not to be the case for the claim that the KKM lemma implies Sperner’s lemma. The goal of this note is to provide such a proof.
MSC:
| 54H25 | Fixed-point and coincidence theorems (topological aspects) |
References:
| [1] | Border, K. C., Fixed Point Theorems with Applications To Economics and Game Theory, (1985), Cambridge University Press Cambridge · Zbl 0558.47038 |
| [2] | Brouwer, L. E.J., Über abbildung von mannigfaltigkeiten, Math. Ann., 71, 1, 97-115, (1911) · JFM 42.0417.01 |
| [3] | Knaster, B.; Kuratowski, C.; Mazurkiewicz, S., Ein beweis des fixpunktsatzes für \(n\)-dimensionale simplexe, Fund. Math., 14, 132-137, (1929) · JFM 55.0972.01 |
| [4] | Sperner, E., Neuer beweis für die invarianz der dimensionszahl und des gebietes, Abh. Math. Semin. Univ. Hambg., 6, 265-272, (1928) · JFM 54.0614.01 |
| [5] | Yoseloff, M., Topological proofs of some combinatorial theorems, J. Combin. Theory Ser. A, 17, 95-111, (1974) · Zbl 0365.05021 |
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