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  "body": "\n## Summary\n\nA Berkovich space is an analytic space over a complete [[non-Archimedean|1H5F1BP6XS]] field whose points are multiplicative seminorms, see the [[definition|3JFF28P5NJ]].\nVladimir Berkovich introduced these spaces as a version of non-Archimedean analytic geometry with good topological properties.\nUnlike the field itself, which is totally disconnected, Berkovich spaces are Hausdorff and locally path-connected, and the spectrum of a Banach algebra is compact.\nThey contain the points of a [[rigid analytic space|YAH6MZVG7K]] and add many new points.\nThis makes them a natural setting for [[non-Archimedean dynamics|6RM4RWWSSX]], potential theory and tropical geometry.\n\n## Explanations\n\n### L1 human\n\nOver the $p$-adic numbers, two discs are either disjoint or one contains the other, and the space breaks into dust.\nBerkovich's idea is to add a new point for every closed disc, together with some limit points.\nA point is described by the way it measures the size of functions.\nOver the field $\\C_p$, the point of a disc measures a polynomial by its largest size on that disc.\nFor the closed unit disc, the new point for the whole disc measures $\\sum a_n x^n$ by $\\max_n \\abs{a_n}$.\nWith these extra points the closed unit disc becomes a connected tree-like space, much closer to the picture of a complex disc.\n",
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