Behnke-Stein theorem

Theorem (Behnke-Stein).

Suppose that $G_k\mathrm{\subset }{\mathrm{C}}^n$ are domains of holomorphysuch that $G_k\mathrm{\subset }{G}_j$ whenever . Then${\mathrm{\bigcup }}_{k\mathrm{=}\mathrm{1}}^{\mathrm{\infty }}{G}_k$ is a domain of holomorphy.

This is related to the fact that an increasing union of pseudoconvex domains is pseudoconvex and so it can be proven using that fact and the solution of theLevi problem. Though historically this theorem was in fact used to solvethe Levi problem.