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

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