Les espaces de Berkovich sont ang\'eliques

TL;DR

This paper demonstrates that a significant portion of the topology of Berkovich spaces can be understood through sequences, establishing their sequential properties despite potential non-metrizability.

Contribution

It proves that Berkovich spaces are angelic, showing limit points are actual limits of sequences and compact sets are sequentially compact, using scalar extension techniques.

Findings

Limit points are actual limits of sequences.

Compact subsets are sequentially compact.

Points can be universally lifted over algebraically closed fields.

Abstract

Although Berkovich spaces may fail to be metrizable when defined over too big a field, we prove that a large part of their topology can be recovered through sequences: for instance, limit points of subsets are actual limits of sequences and compact subsets are sequentially compact. Our proof uses extension of scalars in an essential way and we need to investigate some of its properties. We show that a point in a disc may be defined over a subfield of countable type and that, over algebraically closed fields, every point is universal: in an extension of scalars, it may be canonically lifted.