An increasing sequence of stein manifolds whose limit is not Stein

Abstract

This question was raised in 1933 by Behnke-Thullen [2] in the case when M is an open subset of complex Euclidean space. In the same paper they solved this problem for various special domains M. The problem was solved affirmatively for arbitrary open subsets M in IF" by Behnke-Stein [1, 1938]. K. Stein [4, 1956] proved that M is Stein if each M i is relatively Runge in Mi+ 1. Docquier-Grauert [3, 1960] showed that M is a Stein manifold if there exists a family {Mt},t~[O, 1), of Stein open subsets such that M = U M t and such O ' o_-<t<to Q) Mt=Mt°=int(\to<t<lO Mr). Actually their theorem