Throughout this paper R and S will denote öpen Riemann surfa- ces and X, Y will be non-empty subsets of R and S, respectively. A func- tion 0 : X -> S is said to be analytic if for each point p e X there is an öpen neighborhood Up of p and an analytic function ıpp : Up S such that 4>p and 0 coincide on Up p X. This is eguivalent to assuming that there is a single öpen set Ü a X and an analytic function 0 : U —> S such th at ıp | X = 0.L et A(X,Y) denote the set of ali analytic functions 0 : X S with 0(X ) c Y. For Y = S = C , a functionin A(X, C) is called holomorphic and we write H(X) = A(X,C).