In this paper some
function theoretic properties of an open Riemann surface are related to a
condition on the exhaustion of the surface by finite Riemann surfaces. The
class of Myrberg surfaces is introduced; these are certain branched covering
surfaces of the unit disc. The exhaustion condition is used to distinguish those
Myrberg surfaces on which the bounded analytic functions separate points. A
complete description is given of the ways in which the space of bounded
analytic functions on a Myrberg surface can degenerate. The exhaustion
condition is stated in terms of the Green’s function; it is already known to
be equivalent to a function theoretic condition on the fundamental group
of the surface. This latter condition is shown to imply that the surface is
an open subset of the spectrum of its Banach algebra of bounded analytic
functions.
