Every ultrametric space has the following property: given a pair of intersecting balls, one of the balls will contain the other. Using this fact one can represent the structure of the corresponding coverings of an ultrametric space by a tree or by a chain. Given an ultrametric space \((U,d)\) the corresponding chain consists of partitions \(X_k\) of \(U\) into balls of radius \(2^k\) and the natural inclusions \(\phi_k : X_k \to X_{k+1}\) for \(k\in J\). For every chain considered in the paper one can define an end space as an ultrametric space.
The paper introduces three classifications of ultrametric spaces in various categories using appropriate chains and end spaces. It turns out that the end space is equivalent (in each of the categories below) to the original space \(U\) and that the equivalence of two ultrametric spaces in any of the three categories coincides with the existence of a common zig-zag chain between the corresponding chains.
– The category of complete ultrametric spaces and bi-uniform maps is modeled by chains with \(J\) being the integer set (thus capturing the geometry of large and small coverings).
– The category of complete ultrametric spaces and uniformly maps is modeled by chains with \(J\) essentially being the set of negative integers (thus capturing the geometry of small coverings).
– The category of ultrametric spaces and bornologous multi-maps is modeled by chains with \(J\) being the set of positive integers (thus capturing the geometry of large coverings).