Berkovich space

Summary

A Berkovich space is an analytic space over a complete non-Archimedean field whose points are multiplicative seminorms, see the definition. Vladimir Berkovich introduced these spaces as a version of non-Archimedean analytic geometry with good topological properties. Unlike the field itself, which is totally disconnected, Berkovich spaces are Hausdorff and locally path-connected, and the spectrum of a Banach algebra is compact. They contain the points of a rigid analytic space and add many new points. This makes them a natural setting for non-Archimedean dynamics, potential theory and tropical geometry.