The range of ultrametrics, compactness, and separability

TL;DR

This paper characterizes the structure of range sets of ultrametrics in relation to compactness and separability of ultrametrizable spaces, providing criteria based on order isomorphism and countability of range sets.

Contribution

It establishes a connection between the order type of ultrametric range sets and topological properties like compactness and separability.

Findings

Compact ultrametrizable spaces have order-isomorphic range sets for all compatible ultrametrics.

An ultrametrizable space is compact iff all compatible ultrametrics have order-isomorphic range sets.

Separable ultrametrizable spaces have compatible ultrametrics with at most countable range sets.

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.