# Hausdorff distance between ultrametric balls

## TL;DR

This paper explores the relationship between properties of ultrametric spaces and their closed balls under Hausdorff distance, establishing equivalences for various topological and metric properties.

## Contribution

It provides necessary and sufficient conditions linking properties of an ultrametric space to those of its space of closed balls with Hausdorff metric.

## Key findings

- Equivalence of discreteness, compactness, and other properties between $(X, d)$ and $(ar{ extbf{B}}_X, d_H)$
- Characterization of separability conditions for $(ar{ extbf{B}}_X, d_H)$
- Conditions under which properties like completeness and local compactness are preserved

## 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.