A Mergelyan theorem for mappings to \(\mathbb{C}^2\setminus \mathbb{R}^2\) (Q1307117)

Principal result: Let \(K\) be a compact subset of \(\mathbb{C}\) with \(\mathbb{C} \setminus K\) connected, \(\varepsilon>0\) and \(f\) a continuous map from \(K\) into \(\mathbb{C}^2\setminus \mathbb{R}^2\) which is holomorphic in the interior of \(K\). Then there exists a holomorphic map \(g:\mathbb{C} \to\mathbb{C}^2 \setminus\mathbb{R}^2\) such that \(\|f-g \|_K\leq\varepsilon\). The proof, elementary, is quite nice. The author proves also that the Merguelian theorem remains true for maps \(f:K \to\mathbb{C}^n \setminus A\), where \(A\) is a closed subset of \(\mathbb{C}^n\) with Hausdorff dimension \(<2n-2\).