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Keywords:
point character; uniform cover; continuum hypothesis; Specker graph
Summary:
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.
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