A known Urysohn’s result shows that there exists a universal metric space, i.e., a metric space into every other (separable) metric space can be isomorphically embedded. Moreover, this universal metric space can be selected to be ultra-homogeneous — every isomorphism of its two finite subsets can be extended to the isomorphism of the whole space. Starting with Einstein’s theories of Special and General relativity, space-times are described by a different type of structure — a set (of events) equipped with the proper time τ (a, b) between points a and b; such spaces are known as space-times with kinematic metric, or k-space-times. In this paper, we show that Urysohn’s result can be extended to k-space-times — namely, that there exists an ultra-homogeneous universal k-space-time.
© 2013 Asuman G. Aksoy, Zachary Glassman, Olga M. Kosheleva, Vladik Kreinovich
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A.G. Aksoy, Z. Glassman, O. Kosheleva and V. Kreinovich, From Urysohns universal space to a universal space-time, Mathematical Structures and Modeling, 28(2), pp. 28-34, 2013.