Arakelyan's theorem

In mathematics, Arakelyan's theorem is a generalization of Mergelyan's theorem from compact subsets of an open subset of the complex plane to relatively closed subsets of an open subset.

Let Ω be an open subset of

{\displaystyle \mathbb {C} }

and E a relatively closed subset of Ω.

By Ω* is denoted the Alexandroff compactification of Ω. Arakelyan's theorem states that for every f continuous in E and holomorphic in the interior of E and for every ε > 0 there exists g holomorphic in Ω such that |g − f| < ε on E if and only if Ω* \ E is connected and locally connected.