Can someone check my proof of Liouville's theorem with Riemann surfaces? Let f be a holomorphic bounded function on $\mathbb{C}$. Then using the transition map z to 1/z, we transport this function to a function defined in a neighborhood of the point at infinity (which is 0 now). It's bounded in a neighborhood of 0, so we can holomorphically extend it over 0. We now have a holomorphic function on the entire sphere, so it must be constant, because the sphere is compact.
(the last bit is by the maximum modulus principle for riemann surfaces)

@tb Yesterday, while walking back home, I realised something trivial but interesting (at least, the first time you realise it): about describing something from above and below. This is in connection with our chat with Benjamin. I was thinking about the topological closure operator, and first I thought it can only be defined from above; it didn't have a description from below.

It took a few minutes to realise that sequential characterisation is the right "below below" definition. [I know this works best in metric spaces, and needs additional work for general topologies, but ignore that.]