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Treballs finals del Màster en Matemàtica Avançada, Facultat de Matemàtiques, Universitat de Barcelona: Curs: 2023-2024.  Director: Martín Sombra

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In this Master Final Project I have studied Berkovich spaces, which is one of the existing approaches to non-Archimedean geometry, a branch that deals with analytic spaces over non-Archimedean fields. Let us first give some context on $p$-adic geometry and the necessity to develop such a theory of analytic spaces. 
Any norm gives rise to a metric space by setting the distance between two elements as the norm of their difference. In the case of a metric space induced by a non-Archimedean norm, the topological space is totally disconnected. For this reason, when we try to develop a theory of analytic functions similar that for the
complex case (i.e., the Archimedean case), we encounter some notorious problems.