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Metrization in small and large scale structures

Miyata T.

Bulletin of the Polish Academy of Sciences. Mathematics

2017

Vol. 65, no. 1

81--92

Given a topological structure and a coarse structure on a set, N. Wright gave a necessary and sufficient condition for the set to have a metric inducing simultaneously both the structures. We use the idea of the Alexandroff and Urysohn metrization theorem for topological spaces, to investigate a simultaneous metrization problem for a set with a uniform (and topological) structure and a coarse structure. In particular, we prove that given two metrics dU and dC on a set X such that the uniform (topological) structure induced by dU is compatible in some sense with the coarse structure induced by dC, there exists a metric d on X which is isometric to dU in a small scale and to dC in a large scale. We then apply this idea to show that if, in addition, the uniform space has uniform dimension 0 and the coarse space has asymptotic dimension 0, then there exists an ultrametric d on X which is isometric to dU in small scale and to dC in large scale.