Let $\Omega$ be a connected open set in the complex plane. What is the closure of the polynomials in $\mathcal{H}(\Omega)$ the set of holomorphic functions on $\Omega$? The topology is the usual compact convergence topology. Take, for instance, an annulus such as $D(r,R)$, the set of all complex $z$ such that $r<|z|< R$, you cannot recover the function $z\mapsto \frac{1}{z}$ because of the residue at $0$, so what holomorphic functions are limits of polynomials?



What holomorphic functions are limits of polynomials?
Let $\Sigma\supset\Omega$ be the union of $\Omega$ and all bounded components of ${\mathbb C}\setminus \Omega$. The algebra you get is the algebra of all holomorphic functions on $\Sigma$.


First, every $f\in{\cal H}(\Sigma)$ is a locally uniform limit of polynomials as a consequence of Runge's Theorem, see Corollary 1.15 in John Conway's "Functions of one complex variable I", 2nd Ed.


Second, if $p\_n$ is a sequence of polynomials converging locally uniformly on $\Omega$ and if $K$ is a bounded component of ${\mathbb C}\setminus \Omega$, then $p\_n$ also converges uniformly on a neighborhood of $K$ by Cauchy's integral Theorem, as there exists a path around $K$ with winding number 1. (Take the positively oriented boundary of an $\varepsilon$-neighborhood of $K$, where $\varepsilon$ is chosen so small that it does not hit any other components.)


