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Berkovich spaces are angelic. (Les espaces de Berkovich sont angéliques.) (French. English summary) Zbl 1314.14046
Berkovich spaces exhibit a large number of remarkable topological properties. However, they may fail to be metrizable when defined over too big a field. In the present paper, the author shows that even in non-metrizable situations, a large part of the topological properties of a Berkovich space can be recovered in terms of sequences (in place of nets). More precisely speaking, he shows that Berkovich spaces are Fréchet-Urysohn spaces: every point in the closure of a subset of a Berkovich space is the limit of a sequence of points in that subset. This main result of the present paper plays a key role for the development of a general theory of nonarchimedean dynamical systems. The author also proves that certain Berkovich cases (like, for example, curves or algebraizations of algebraic varieties) are angelic, meaning that their relatively \(\omega\)-compact subsets are relatively compact.
The proofs rely on an extension of scalar techniques: the author shows that every point in a disc can be defined over a subfield of countable type. Moreover, he shows that over algebraically closed fields, every point of a Berkovich space may be canonically lifted to spaces obtained via extension of scalars. These two results combined allow to descend a given Berkovich space to a metrizable space and then lift the Fréchet-Urysohn property back to the original space. To obtain them, the author generalizes a result of Berkovich on Shilov boundaries from a strictly affinoid to a general affinoid setting, using graded reduction techniques as they have been developed by M. Temkin [Isr. J. Math. 140, 1–27 (2004; Zbl 1066.32025)]. Along these lines, he generalizes some classical results from commutative algebra (such as the Nullstellensatz) to the graded setting.

14G22 Rigid analytic geometry
54D55 Sequential spaces
46A50 Compactness in topological linear spaces; angelic spaces, etc.
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