For any ultrametric space, its set of closed balls with the Hausdorff distance inherits discreteness, local finiteness, completeness, compactness, and related properties exactly when the original space has them; separability of the ball space is equivalent to countability of the positive-radius ball
Compact ultrametric spaces generated by labeled star graphs
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abstract
Let US be the class of all ultrametric spaces generated by labeled star graphs. We prove that compact US-spaces are the completions of totally bounded ultrametric spaces generated by decreasingly labeled rays. We characterize the ultrametric spaces which are weakly similar to finite US-spaces and describe these spaces by certain four-point conditions.
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Hausdorff distance between ultrametric balls
For any ultrametric space, its set of closed balls with the Hausdorff distance inherits discreteness, local finiteness, completeness, compactness, and related properties exactly when the original space has them; separability of the ball space is equivalent to countability of the positive-radius ball