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arXiv · 2102.10901

The range of ultrametrics, compactness, and separability

Oleksiy Dovgoshey·Volodymir Shcherbak
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Abstract

We describe the order type of range sets of compact ultrametrics and show that an ultrametrizable infinite topological space $(X, \tau)$ is compact iff the range sets are order isomorphic for any two ultrametrics compatible with the topology $\tau$. 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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Oleksiy Dovgoshey, Volodymir Shcherbak. 2021-02-22. The range of ultrametrics, compactness, and separability. https://arxiv.org/abs/2102.10901

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