Metrization in small and large scale structures
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 $d_U$ and $d_C$ on a set $X$ such that the uniform (topological) structure induced by $d_U$ is compatible in some sense with the coarse structure induced by $d_C$, there exists a metric $d$ on $X$ which is isometric to $d_U$ in a small scale and to $d_C$ 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 $d_U$ in small scale and to $d_C$ in large scale.