Equivalent characterizations of non-Archimedean uniform spaces

TL;DR

This paper explores various equivalent ways to characterize non-Archimedean uniform spaces, linking their definitions via diagonal uniformities, covering uniformities, and pseudo-metrics, and provides conditions for when such spaces induce specific topologies.

Contribution

It establishes the equivalence of different non-Archimedean uniformity concepts and characterizes when these uniformities induce particular topologies or come from a single pseudo-metric.

Findings

Non-Archimedean uniformities are equivalent across multiple definitions.

A separation axiom characterizes when a topology is induced by a non-Archimedean uniformity.

Conditions are provided for a non-Archimedean uniformity to be metrizable.

Abstract

In this paper, we deal with uniform spaces whose diagonal uniformity admits a basis consisting of equivalence relations. Such non-Archimedean uniform spaces are particularly interesting for applications in commutative ring theory, because uniformities stemming from valuations or directed systems of ideals are of this type. In general, apart from diagonal uniformities, there are two further approaches to the concept of uniform spaces: covering uniformities and systems of pseudo-metrics. For each of these ways of defining a uniformity, we isolate a non-Archimedean special case and show that these special cases themselves are equivalent to the notion of non-Archimedean diagonal uniformities. Moreover, we formulate a seperation axiom for topological spaces that tells exactly when the topology is induced by a non-Archimedean uniformity. In analogy to the classical metrizability theorems, we characterize when a non-Archimedean uniformity comes from a single pseudo-metric.