Corollary 1

The following conditions are equivalent:

(i) The set E meets all the bounded connected components of $\mathbf{C}-$ X ;

(ii) The mapping $\chi \mapsto \chi (z)$ is a homeomorphism of $\mathsf{X}(R_E(X))$ onto X ;

(iii) One has $R_E(X) = R(X)$.

Let $E'=\mathbf{C}-$ X. The conditions (i), (ii) and (iii) are respectively equivalent to $I(E) = I(E')$, to $X_E= X_{E'}$ (by prop. 5, d)) and to $R_E(X) = R_{E'}(X)$. Hence they are equivalent among themselves by prop. 5, e).

The following corollary makes the theorem of Runge more precise (th. 3 of I, p. 69).