Radó's theorem (Riemann surfaces)

In mathematical complex analysis, Radó's theorem, proved by Tibor Radó (1925), states that every connected Riemann surface is second-countable (has a countable base for its topology).

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.

References

• Hubbard, John Hamal (2006), Teichmüller theory and applications to geometry, topology, and dynamics. Vol. 1, Matrix Editions, Ithaca, NY, ISBN 978-0-9715766-2-9, MR 2245223
• Radó, Tibor (1925), "Über den Begriff der Riemannschen Fläche", Acta Szeged, 2 (2): 101–121, JFM 51.0273.01