1957/12/01 by Maurice Heins · 1 citation
Mathematics · #History and Theory of Mathematics #Geometric Analysis and Curvature Flows #Mathematics #Countable set #Riemann surface #Uniformization theorem #Geometric function theory #Pure mathematics #Conformal map #Riemann sphere #Riemann hypothesis #Fundamental theorem #Simple (philosophy) #Connection (principal bundle) #Discrete mathematics #Mathematical analysis #Riemann–Hurwitz formula #Geometry #Fixed-point theorem
paper · pdf · doi:10.1017/s002776300002198x
openalex publication_date 1957/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/04
On reviewing recently the proof which I gave for the Riemann mapping theorem for simply-connected Riemann surfaces several years ago [2], I observed that the argument which I used could be so modified that the assumption of a countable base could be completely eliminated. The problem of treating the Riemann mapping theorem without this assumption has been current for some time. The object of the present note is to give an account of a solution of this question. Of course, the classical theorem of Radó permits us to dispense with an attack on the Riemann mapping theorem which does not appeal to the countable base assumption. In this connection, we recall that Nevanlinna [4] has given a straightforward potential-theoretic treatment of the Radó theorem in which neither the Riemann mapping theorem (nor the notion of a universal covering) enters as they do in Radó’s proof. Nevertheless, a certain technical interest attaches to a direct treatment of the Riemann mapping theorem without the countable base assumption. An immediate byproduct of such a treatment is a simple proof of the Radó theorem which invokes the notion of a universal covering but in a manner different from that of Radó’s proof. Indeed, it suffices to note that a manifold has a countable base if the domain of a universal covering does.