Separation and weak separation on Riemann surfaces (Q1277029)

Let \(R\) be a Riemann surface, and let \(A\) be an algebra of analytic functions on \(R\). \(A\) weakly separates the points of \(R\) if there is a discrete subset \(\Lambda \) of \(R\) such that \(A\) separates the points of \(R\setminus \Lambda \). It is shown that \(A\) weakly separates the points of \(R\) if and only if there exists a sequence of relatively compact open sets \((D_n)\) in \(R\) such that (i) \(\partial D_n \) is connected, (ii) \(\overline{D_1} \subset \overline{D_2} \subset \dots \), (iii) \(R=\bigcup \overline{D_n}\), and (iv) \(A\) separates the points of a neighborhood of \(\partial D_n\). It is also pointed out that condition (i) cannot be removed and also that ''a neighborhood of'' cannot be removed in condition (iv).