Hausdorff distance between ultrametric balls

Abstract

Let $(X,d)$ be an ultrametric space and let $d_H$ be the Hausdorff distance on the set ${\bar{\mathbf{B}}}_X$ of all closed balls in $(X,d)$. Some interconnections between the properties of the spaces $(X,d)$ and $({\bar{\mathbf{B}}}_X,d_H)$ are described. It is established that the space $({\bar{\mathbf{B}}}_X,d_H)$ has such properties as discreteness, local finiteness, metrical discreteness, completeness, compactness, local compactness if and only if the space $(X,d)$ has these properties. Necessary and sufficient conditions for the separability of the space $({\bar{\mathbf{B}}}_X,d_H)$ are also proved.

arXiv comment28 pages