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.
Theorem
Let Ω be an open subset of 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.
References
Arakelyan's theorem Wikipedia(Text) CC BY-SA