On the harmonic boundary of an open Riemann surface, I

Every open Riemann surface R can be represented as a dense subset of a compact Hausdorff space R* considered by Gelfand and Royden (cf. [3], [4], [10]). The ideal boundary of R is then realized as a closed non-dense subset ƒ¡ of R*, which is, by definition, the set of maximal ideals containing the ideal K of BD functions with compact carriers. The set ‡TM of maximal ideals containing K (closure of K by BD-topology) constitutes an essential part of ƒ¡ and is called the harmonic boundary of R. In this article we shall study some properties on harmonic boundaries and non compact subregions on R. By a non-compact subregion G on R we shall understand in the sequel a non-compact domain on R whose relative boundary •ÝG consists of an at most countable number of disjoint analytic Jordan curves not accumulating to any point of R.