Behnke-Stein theorem

The union of domains of holomorphy $G_k \subset \mathbf{C}^n$, $k=1,2,\ldots$ for which $G_k \subseteq G_{k+1}$ for all $k$, is also a domain of holomorphy. The Behnke–Stein theorem is valid not only in the complex Euclidean space $\mathbf{C}^n$, but also on any Stein manifold. If the sequence $G_k$ is not monotone increasing under imbedding, the theorem is not valid; e.g. the union of the two domains of holomorphy $$ G_1 = \{ (z_1,z_2) : |z_1| <1 \,,\ |z_2| < 2 \} $$ and $$ G_2 = \{ (z_1,z_2) : |z_1| <2 \,,\ |z_2| < 1 \} $$ in $\mathbf{C}^2$ is not a domain of holomorphy.

References

[1]	H. Behnke, K. Stein, "Konvergente Folgen von Regularitätsbereichen und die Meromorphiekonvexität" Math. Ann. , 116 (1938) pp. 204–216 DOI 10.1007/BF01597355 Zbl 0020.37803
[2]	V.S. Vladimirov, "Methods of the theory of functions of several complex variables" , M.I.T. (1966) (Translated from Russian)
[a1]	Hans Grauert, Reinhold Remmert, "Theory of Stein spaces" (Tr. Alan Huckleberry) Grundlehren der mathematischen Wissenschaften 236 Springer (1979, repr.2008) ISBN 3-540-00373-8 Zbl 0433.32007