Prüfer surface

An example of a two-dimensional real-analytic manifold (cf. also Analytic manifold) not having a countable basis of open sets.

It was introduced in a paper of T. Radó [1]. There is a generalization of the Prüfer surface to any even dimension (cf. [2]).

However, every Riemann surface has a countable basis of open sets (Radó's theorem).

References

• [1] T. Radó, "Ueber den Begriff der Riemannschen Flächen" Acta Szeged , 2 (1925) pp. 101–121
• [2] E. Calabi, M. Rosenlicht, "Complex analytic manifolds without countable base" Proc. Amer. Math. Soc. , 4 (1953) pp. 335–340
• [3] G. Springer, "Introduction to Riemann surfaces" , Addison-Wesley (1957) Chap. 10
• [4] R. Nevanlinna, "Uniformisierung" , Springer (1953)