Let $X$ be a Riemann Surface and $K$ a compact subset of $X$. Every holomorphic function in $K$ be uniformly approximable on $K$ by holomorphic functions on $X$ if $X-K$ have no connected component with compact closure in $X$.
Is the conversely also true?
Yes .Suppose U is a relatively compact component of the complement of K and p a point in U.The frontier of U lies in K .By Weierstrass Let g be a holomorphic function that vanishes at p and nowhere else on X and f the reciprocal of g.By assumption we can find a sequence of holomorphic functions on X converging uniformly on K to f .Now this sequence multiplied by g converges to one uniformly on the frontier of U and by the maximum principle uniformly to one on U .This contradicts the fact every term of the sequence multiplied by g vanishes at p . Therefore U cannot be relatively compact .
