Arakelyan's theorem

Remarks

For compact $E$, the set $\hat\C \setminus E$ is an open subset of the sphere, so it is automatically locally connected, and connectedness of $\hat\C \setminus E$ is the same as connectedness of $\C \setminus E$. So for compact $E$ the theorem reduces to Mergelyan's theorem, since entire functions can be approximated by polynomials on compact sets. The new condition for unbounded $E$ is local connectedness of $\hat\C \setminus E$ at the point $\infty$.

Example: let $E = \R$. Then $\hat\C \setminus \R$ is the union of the two open half-planes and the point $\infty$, which is connected and locally connected. So every continuous function on $\R$ is a uniform limit on all of $\R$ of entire functions, a result first proved by Carleman.

Arakelyan also proved a version on an arbitrary open set $\Omega$ in place of $\C$, with $E$ closed in $\Omega$. There the approximants are holomorphic on $\Omega$, and the one-point compactification of $\Omega$ replaces $\hat\C$.

Going deeper

For a student

The theorem is the final answer to uniform holomorphic approximation on closed subsets of the plane. It is used to construct entire functions with prescribed behaviour along unbounded sets, for example functions that tend to $0$ along every ray but are not identically $0$. Its proof combines Mergelyan-type estimates with a careful exhaustion of $E$ by compact pieces, where local connectedness at $\infty$ controls how the pieces fit together.