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Keywords:
simplex; colouring; covering dimension; point-finite; fixed point; algebraic topology
Summary:
We prove an extension of the well-known combinatorial-topological lemma of E. Sperner to the case of infinite-dimensional cubes. It is obtained as a corollary to an infinitary extension of the Lebesgue Covering Dimension Theorem.
References:
[1] Benyamini Y., Sternfeld Y.: Spheres in infinite-dimensional normed spaces are Lipschitz contractible. Proc. Amer. Math. Soc. 88:3 (1983), 439-445. MR 0699410 | Zbl 0518.46010
[2] Brown A.B., Cairns S.: Strengthening of Sperner's lemma applied to homology theory. Proc. Nat. Acad. Sci. U.S.A. 47 (1961), 113-114. MR 0146831 | Zbl 0097.38702
[3] Cohen D.I.A.: On the Sperner lemma. J. Combinatorial Theory 2 (1967), 585-587. MR 0214047 | Zbl 0163.18104
[4] Engelking R.: Dimension Theory. Polish Scientific Publishers, Warszawa, 1978. MR 0482697 | Zbl 0401.54029
[5] Erdös P., Galvin F., Hajnal A.: On set-systems having large chromatic number and not containing prescribed subsystems. in: Infinite and Finite Sets (A. Hajnal, R. Rado, V.T. Sós, Eds.), North-Holland, Amsterdam, 1975, pp.425-513. MR 0398876
[6] Fan Ky: A generalization of Tucker's combinatorial lemma with topological applications. Ann. of Math. (2) 56 (1952), 431-437. MR 0051506 | Zbl 0047.42004
[7] Fried J.: Personal communication.
[8] Goebel K.: On the minimal displacement of points under Lipschitzian mappings. Pacific J. Math. 45 (1973), 151-163. MR 0328708 | Zbl 0265.47046
[9] Isbell J.R.: Uniform Spaces. Mathematical Surveys 12, Amer. Math. Soc., Providence, Rhode Island, 1964. MR 0170323 | Zbl 0124.15601
[10] Knaster B., Kuratowski C., Mazurkiewicz S.: Ein Beweis des Fixpunksatzes für $n$-dimensionale Simplexe. Fund. Math. 14 (1929), 132-137.
[11] Kryński S.: Remarks on matroids and Sperner's lemma. European J. Combin. 11 (1990), 485-488. MR 1075536 | Zbl 0727.05015
[12] Kuhn H.W.: Some combinatorial lemmas in topology. IBM J. Res. Develop. 4 (1960), 508-524. MR 0124038 | Zbl 0109.15603
[13] Lindström S.: On matroids and Sperner's lemma. European J. Combin. 2 (1981), 65-66. MR 0611933 | Zbl 0473.05022
[14] Lóvasz L.: Matroids and Sperner's lemma. European J. Combin. 1 (1980), 65-66. MR 0576768 | Zbl 0443.05025
[15] Mani P.: Zwei kombinatorisch-geometrische Sätze vom Typus Sperner-Tucker-Ky Fan. Monatsh. Math. 71 (1967), 427-435. MR 0227859 | Zbl 0173.26202
[16] Pelant J.: Combinatorial properties of uniformities. General Topology and its Relations to Modern Analysis and Algebra IV, Lecture Notes in Mathematics 609, Springer, Berlin-Heidelberg-New York, 1977, pp.154-165. MR 0500846 | Zbl 0371.54054
[17] Pelant J.: Embeddings into $c_0$. Topology Appl. 57 (1994), 2-3 259-269. MR 1278027
[18] Pelant J., Rödl V.: On coverings of infinite-dimensional metric spaces. Topological, algebraical and combinatorial structures. Frolík's memorial volume. Discrete Math. 108 (1992), 1-3 75-81. MR 1189831
[19] Rödl V.: Small spaces with large point-character. European J. Combin. 8 (1987), 55-58. MR 0884064
[20] Smith J.C.: Characterizations of metric-dependent dimension functions. Proc. Amer. Math. Soc. 19:6 (1968), 1264-1269. MR 0232365 | Zbl 0169.25103
[21] Sperner E.: Neuer Beweis für die Invarianz der Dimensionzahl und des Gebietes. Abh. Math. Sem. Hamburg 6 (1928), 265-272.
[22] Sperner E.: Kombinatorik bewerter Komplexe. Abh. Math. Sem. Univ. Hamburg 39 (1973), 21-43. MR 0332498
[23] Stone A.H.: Universal spaces for some metrizable uniformities. Quart. J. Math. Oxford, Ser. (2) 11 (1960), 105-115. MR 0116308 | Zbl 0096.37402
[24] Ščepin E.V.: On a problem of Isbell. Soviet Math. Dokl. 16 (1975), 685-687. MR 0380743
[25] Tucker A.W.: Some topological properties of disk and sphere. in: Proc. First Canadian Math. Congress, Montreal, Canada, 1945, pp.285-309. MR 0020254 | Zbl 0061.40305
[26] Vidossich G.: Uniform spaces of countable type. Proc. Amer. Math. Soc. 25:3 (1970), 551-553. MR 0261546 | Zbl 0181.50903
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