Uniform approximation by meromorphic functions on Riemann surfaces

A closed subset E of a Riemann surface S is called a set of uniform meromorphic approximation if every function f continuous on E and holomorphic on EO can be approximated uniformly on E by meromorphic functions on S. We show that if E is a set of uniform meromorphic approximation, then so is E boolean AND (D) over bar for every compact parametric disk A As a consequence, we obtain a generalization to Riemarm surfaces of a well-known theorem of A. G. Vitushkin.