Simultaneous approximation on two subsets of an open Riemann surface (Q818467)

A closed subset \(E\) of an open Riemann surface \(R\) is called a holomorphic (meromorphic) approximation set iff each function holomorphic (meromorphic) on an open neighbourhood of \(E\) can be uniformly approximated by functions holomorphic (meromorphic) on \(R\). The problem of characterizing these approximation sets is solved for Riemann surfaces of finite genus -- Theorems of Runge, Scheinberg, Roth, Arakelian and other authors. The authors start with a recapitulation of the known results and present many new results for the case of Riemann surfaces of infinite genus.