View source for Behnke-Stein theorem

The union of [[domain of holomorphy |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====
<table>
<TR><TD valign="top">[1]</TD> <TD valign="top"> 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}}</TD></TR>
<TR><TD valign="top">[2]</TD> <TD valign="top"> V.S. Vladimirov, "Methods of the theory of functions of several complex variables" , M.I.T.  (1966)  (Translated from Russian)</TD></TR>
<TR><TD valign="top">[a1]</TD> <TD valign="top"> 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}}
</table>

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