Mergelyan's theorem

Mergelyan's theorem also holds for open Riemann surfaces.

Then: Mergelyan's theorem does not always hold in higher dimensions (spaces of several complex variables), but it has some consequences.

is not connected, in the initial approximation problem the polynomials have to be replaced by rational functions.

An important step of the solution of this further rational approximation problem was also suggested by Mergelyan in 1952.

Further deep results on rational approximation are due to, in particular, A. G. Vitushkin.

After Weierstrass and Runge, many mathematicians (in particular Walsh, Keldysh, Lavrentyev, Hartogs, and Rosenthal) had been working on the same problem.