The Behnke-Stein theorem for open Riemann surfaces

Abstract:

Using the Riemann-Roch theorem and the set-topological part of Bishop’s special polyhedron lemma, we show that the usual Runge approximation theorem for compact subsets of the Riemann sphere is valid word-for-word on any compact Riemann surface $X$, with meromorphic functions on $X$ playing the role of rational functions; this result is essentially equivalent to the Behnke-Stein approximation theorem.