Equivalent characterizations of non-Archimedean uniform spaces

In this paper, we deal with uniform spaces given by a system of non-Archimedean pseudo-metrics. 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 systems of pseudo-metrics, there are two further approaches to the concept of uniform spaces: covering uniformities and diagonal uniformities. For each of these ways of defining a uniformity, we isolate a non-Archimedean special case and show that these special cases themselves correspond to systems of non-Archimedean pseudo-metrics.

Moreover, we formulate a separation axiom that tells exactly when a 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.