Rigid p-adic geometry and Berkovich spaces

I'll define some algebras and geometric spaces that aim to mimic the closed unit disk and its ring of analytic functions in the non-archimedean world. Unfortunately, like most topological spaces that arise in the $p$-adic universe, these spaces are totally disconnected. To remedy this problem, I will tell you what Berkovich spaces are and draw some pictures of uniquely path-connected Hausdorff spaces. We will pretend that these pictures look like the closed unit disk in the complex plane.