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I determine the relationship between $\mathbb{C}_p^\times \times\mathbb{C}_p$ and a Berkovich space via an equivalence relation obtained bycoarse-graining. This process also establishes a correspondence between$p$-adic fields and Bruhat-Tits trees. From this space, I give a previouslyunknown construction of the Berkovich projective line$\mathbb{P}_\text{Berk}^1$ and outline a direction for the development of acomprehensive theory of non-Archimedean AdS/CFT and consequently, adelicAdS/CFT, enabled by this construction.

On $\mathbb{P}_{Berk}^1$ and Coarse-Grainings of Non-Archimedean Product Spaces