C. Avart, P. Komjath, V. Rödl
Metric spaces with point character equal to their size

Comment.Math.Univ.Carolin. 51,3 (2010) 459-467.

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, {\it 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 R\"odl V., {\it Small spaces with large point character\/}, European J. Combin. {\bf 8} (1987), no.~1, 55--58. The goal of this note is to remove this assumption.

Keywords: point character, uniform cover, continuum hypothesis, Specker graph.
AMS Subject Classification: 05C12 05C15 54A99 54A25 03E05