An infinitary version of Sperner's Lemma

Aarno Hohti

Commentationes Mathematicae Universitatis Carolinae (2006)

  • Volume: 47, Issue: 3, page 503-514
  • ISSN: 0010-2628

Abstract

top
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.

How to cite

top

Hohti, Aarno. "An infinitary version of Sperner's Lemma." Commentationes Mathematicae Universitatis Carolinae 47.3 (2006): 503-514. <http://eudml.org/doc/249850>.

@article{Hohti2006,
abstract = {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.},
author = {Hohti, Aarno},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {simplex; colouring; covering dimension; point-finite; fixed point; algebraic topology; simplex; colouring; covering dimension; point-finite; fixed point; algebraic topology},
language = {eng},
number = {3},
pages = {503-514},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {An infinitary version of Sperner's Lemma},
url = {http://eudml.org/doc/249850},
volume = {47},
year = {2006},
}

TY - JOUR
AU - Hohti, Aarno
TI - An infinitary version of Sperner's Lemma
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2006
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 47
IS - 3
SP - 503
EP - 514
AB - 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.
LA - eng
KW - simplex; colouring; covering dimension; point-finite; fixed point; algebraic topology; simplex; colouring; covering dimension; point-finite; fixed point; algebraic topology
UR - http://eudml.org/doc/249850
ER -

References

top
  1. Benyamini Y., Sternfeld Y., Spheres in infinite-dimensional normed spaces are Lipschitz contractible, Proc. Amer. Math. Soc. 88:3 (1983), 439-445. (1983) Zbl0518.46010MR0699410
  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. (1961) Zbl0097.38702MR0146831
  3. Cohen D.I.A., On the Sperner lemma, J. Combinatorial Theory 2 (1967), 585-587. (1967) Zbl0163.18104MR0214047
  4. Engelking R., Dimension Theory, Polish Scientific Publishers, Warszawa, 1978. Zbl0401.54029MR0482697
  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. MR0398876
  6. Fan Ky, A generalization of Tucker's combinatorial lemma with topological applications, Ann. of Math. (2) 56 (1952), 431-437. (1952) Zbl0047.42004MR0051506
  7. Fried J., Personal communication, . 
  8. Goebel K., On the minimal displacement of points under Lipschitzian mappings, Pacific J. Math. 45 (1973), 151-163. (1973) Zbl0265.47046MR0328708
  9. Isbell J.R., Uniform Spaces, Mathematical Surveys 12, Amer. Math. Soc., Providence, Rhode Island, 1964. Zbl0124.15601MR0170323
  10. Knaster B., Kuratowski C., Mazurkiewicz S., Ein Beweis des Fixpunksatzes für n -dimensionale Simplexe, Fund. Math. 14 (1929), 132-137. (1929) 
  11. Kryński S., Remarks on matroids and Sperner's lemma, European J. Combin. 11 (1990), 485-488. (1990) Zbl0727.05015MR1075536
  12. Kuhn H.W., Some combinatorial lemmas in topology, IBM J. Res. Develop. 4 (1960), 508-524. (1960) Zbl0109.15603MR0124038
  13. Lindström S., On matroids and Sperner's lemma, European J. Combin. 2 (1981), 65-66. (1981) Zbl0473.05022MR0611933
  14. Lóvasz L., Matroids and Sperner's lemma, European J. Combin. 1 (1980), 65-66. (1980) Zbl0443.05025MR0576768
  15. Mani P., Zwei kombinatorisch-geometrische Sätze vom Typus Sperner-Tucker-Ky Fan, Monatsh. Math. 71 (1967), 427-435. (1967) Zbl0173.26202MR0227859
  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. Zbl0371.54054MR0500846
  17. Pelant J., Embeddings into c 0 , Topology Appl. 57 (1994), 2-3 259-269. (1994) MR1278027
  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. (1992) MR1189831
  19. Rödl V., Small spaces with large point-character, European J. Combin. 8 (1987), 55-58. (1987) MR0884064
  20. Smith J.C., Characterizations of metric-dependent dimension functions, Proc. Amer. Math. Soc. 19:6 (1968), 1264-1269. (1968) Zbl0169.25103MR0232365
  21. Sperner E., Neuer Beweis für die Invarianz der Dimensionzahl und des Gebietes, Abh. Math. Sem. Hamburg 6 (1928), 265-272. (1928) 
  22. Sperner E., Kombinatorik bewerter Komplexe, Abh. Math. Sem. Univ. Hamburg 39 (1973), 21-43. (1973) MR0332498
  23. Stone A.H., Universal spaces for some metrizable uniformities, Quart. J. Math. Oxford, Ser. (2) 11 (1960), 105-115. (1960) Zbl0096.37402MR0116308
  24. Ščepin E.V., On a problem of Isbell, Soviet Math. Dokl. 16 (1975), 685-687. (1975) MR0380743
  25. Tucker A.W., Some topological properties of disk and sphere, in: Proc. First Canadian Math. Congress, Montreal, Canada, 1945, pp.285-309. Zbl0061.40305MR0020254
  26. Vidossich G., Uniform spaces of countable type, Proc. Amer. Math. Soc. 25:3 (1970), 551-553. (1970) Zbl0181.50903MR0261546

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.