Radó's theorem (Riemann surfaces)


In mathematical complex analysis, Radó's theorem, proved by, states that every connected Riemann surface is second-countable.
The Prüfer surface is an example of a surface with no countable base for the topology, so cannot have the structure of a Riemann surface.
The obvious analogue of Radó's theorem in higher dimensions is false: there are 2-dimensional connected complex manifolds that are not second-countable.