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) pp. Chapt.10
[4]	R. Nevanlinna, "Uniformisierung" , Springer (1953)