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I'm trying to understand if/how nonarchimedean uniformities generalize valued fields and ultrametric spaces. It is clear that a nonarchimedean pseudometric induces a nonarchimedean uniformity. Under what conditions is the reverse implication true?

It is known that a space with a nonarchimedean uniformity is necessarily pseudometrizable by a nonarchimedean pseudometric. When are all (pseudo)metrics compatible with the nonarchimedean uniformity necessarily nonarchimedean?

I did not read the whole of A C M van Rooij's book, just the paper titled "Non-Archimedean Uniformities", so perhaps I'm missing the right reference.

Thanks to Aaron Liu, there is a clear counterexample of a space with a nonarchimedean uniformity but not ultrametric: consider the set $ X :\{0,1,2\} $ where $ d(x,y) = \| x - y \| $. This has a discrete topology and a discrete uniformity, but is not ultrametric: $ d(0, 2) > \max(d(0, 1), d(1, 2)) $. So this discrete topology counterexample also prevents considering the "strongly zero dimensional iff ultrametrizable" for metrizable spaces from being relevant here.

Yakov Pechersky
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  • The first linked paper has been published (but keep the ArXiV link !): Windisch, Daniel. Equivalent characterizations of non-Archimedean uniform spaces. Algebraic, number theoretic, and topological aspects of ring theory, 463--474, Springer, Cham, [2023], ©2023. MR4647309 – J.-E. Pin May 01 '25 at 08:37

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