Metric spaces with point character equal to their size

C. Avart; P. Komjath; Vojtěch Rödl

Commentationes Mathematicae Universitatis Carolinae (2010)

  • Volume: 51, Issue: 3, page 459-467
  • ISSN: 0010-2628

Abstract

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In this paper we consider the point character of metric spaces. This parameter which is a uniform version of dimension, was introduced in the context of uniform spaces in the late seventies by Jan Pelant, Cardinal reflections and point-character of uniformities, Seminar Uniform Spaces (Prague, 1973–1974), Math. Inst. Czech. Acad. Sci., Prague, 1975, pp. 149–158. Here we prove for each cardinal κ , the existence of a metric space of cardinality and point character κ . Since the point character can never exceed the cardinality of a metric space this gives the construction of metric spaces with “largest possible” point character. The existence of such spaces was already proved using GCH in Rödl V., Small spaces with large point character, European J. Combin. 8 (1987), no. 1, 55–58. The goal of this note is to remove this assumption.

How to cite

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Avart, C., Komjath, P., and Rödl, Vojtěch. "Metric spaces with point character equal to their size." Commentationes Mathematicae Universitatis Carolinae 51.3 (2010): 459-467. <http://eudml.org/doc/38142>.

@article{Avart2010,
abstract = {In this paper we consider the point character of metric spaces. This parameter which is a uniform version of dimension, was introduced in the context of uniform spaces in the late seventies by Jan Pelant, Cardinal reflections and point-character of uniformities, Seminar Uniform Spaces (Prague, 1973–1974), Math. Inst. Czech. Acad. Sci., Prague, 1975, pp. 149–158. Here we prove for each cardinal $\kappa $, the existence of a metric space of cardinality and point character $\kappa $. Since the point character can never exceed the cardinality of a metric space this gives the construction of metric spaces with “largest possible” point character. The existence of such spaces was already proved using GCH in Rödl V., Small spaces with large point character, European J. Combin. 8 (1987), no. 1, 55–58. The goal of this note is to remove this assumption.},
author = {Avart, C., Komjath, P., Rödl, Vojtěch},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {point character; uniform cover; continuum hypothesis; Specker graph; point character; uniform cover; continuum hypothesis; Specker graph},
language = {eng},
number = {3},
pages = {459-467},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Metric spaces with point character equal to their size},
url = {http://eudml.org/doc/38142},
volume = {51},
year = {2010},
}

TY - JOUR
AU - Avart, C.
AU - Komjath, P.
AU - Rödl, Vojtěch
TI - Metric spaces with point character equal to their size
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2010
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 51
IS - 3
SP - 459
EP - 467
AB - In this paper we consider the point character of metric spaces. This parameter which is a uniform version of dimension, was introduced in the context of uniform spaces in the late seventies by Jan Pelant, Cardinal reflections and point-character of uniformities, Seminar Uniform Spaces (Prague, 1973–1974), Math. Inst. Czech. Acad. Sci., Prague, 1975, pp. 149–158. Here we prove for each cardinal $\kappa $, the existence of a metric space of cardinality and point character $\kappa $. Since the point character can never exceed the cardinality of a metric space this gives the construction of metric spaces with “largest possible” point character. The existence of such spaces was already proved using GCH in Rödl V., Small spaces with large point character, European J. Combin. 8 (1987), no. 1, 55–58. The goal of this note is to remove this assumption.
LA - eng
KW - point character; uniform cover; continuum hypothesis; Specker graph; point character; uniform cover; continuum hypothesis; Specker graph
UR - http://eudml.org/doc/38142
ER -

References

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  4. Stone A.H., 10.1090/S0002-9904-1948-09118-2, Bull. Amer. Math. Soc. 54 (1948), 977–982. Zbl0032.31403MR0026802DOI10.1090/S0002-9904-1948-09118-2
  5. Stone A.H., 10.1093/qmath/11.1.105, Quart. J. Math. 11 (1960), 105–115. MR0116308DOI10.1093/qmath/11.1.105
  6. Isbell J.R., Uniform Spaces, Mathematical Surveys, 12, American Mathematical Society, Providence, RI, 1964. Zbl0124.15601MR0170323
  7. J. Pelant, Cardinal reflections and point-character of uniformities, Seminar Uniform Spaces (Prague, 1973–1974), Math. Inst. Czech. Acad. Sci., Prague, 1975, pp. 149–158. Zbl0326.54020MR0445460
  8. J. Pelant, Uniform metric spaces, Seminar Uniform Spaces 1975-1977 directed by Z. Frolík, Math. Inst. Czech. Acad. Sci., Prague, 1976, pp. 49–53. 
  9. Pelant J., Rödl V., 10.1016/0012-365X(92)90662-Y, Discrete Math. 108 (1992), no. 1–3, 75–81. MR1189831DOI10.1016/0012-365X(92)90662-Y
  10. Rödl V., Canonical partition relations and point character of ℓ 1 spaces, Seminar Uniform Spaces 1976-1977, pp. 79–81. 
  11. Rödl V., 10.1016/S0195-6698(87)80020-3, European J. Combin. 8 (1987), no. 1, 55–58. MR0884064DOI10.1016/S0195-6698(87)80020-3
  12. Schepin E.V., On a problem of Isbell, Dokl. Akad. Nauk SSSR 222 (1976), 541–543. MR0380743

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