The range of ultrametrics, compactness, and separability

We describe the order type of range sets of compact ultrametrics and show that an ultrametrizable infinite topological space $(X, τ)$ is compact iff the range sets are order isomorphic for any two ultrametrics compatible with the topology $τ$. It is also shown that an ultrametrizable topology is separable iff every compatible with this topology ultrametric has at most countable range set.

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