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
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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