Normal sequences in all scales

The metrization theorem of Alexandroff and Urysohn is known as a strong tool in general topology. In this paper, using the notion of Alexandroff and Urysohn metric, we introduce the notions of all-scale normal sequence, large-scale normal sequence, and small-scale normal sequence to establish a new unified approach to study various concepts of metric geometry in all scales. This approach provides us with the interplay between topology and metric geometry. We show that all-scale, large-scale, and small-scale concepts of metric spaces can be studied as topological problems, using the notions of normal sequences in appropriate scales. The concepts which we deal with are metrization, Lipschitz category, the Assouad–Nagata dimension as an all-scale concept; the uniform category, the topological category, the uniform dimension, the covering dimension as a small-scale concept; the coarse category, large-scale Lipschitz maps, quasi-isometric maps, the asymptotic dimension, and the asymptotic Assouad–Nagata dimension as a large-scale concept. This approach also provides us with the interplay between various concepts of metric geometry in different scales. As an application, we obtain some relations of the Assouad–Nagata dimension to the uniform dimension, the covering dimension, and the asymptotic dimension.