Abstract:
We show that if $X$ is a Stein space and, if $\Omega\subset X$ is exhaustable
by a sequence $\Omega_{1}\subset\Omega_{2}\subset\ldots\subset\Omega_{n}\subset\dots$
of open Stein subsets of $X$, then $\Omega$ is Stein. This generalizes a well-known result
of Behnke and Stein which is obtained for $X=\mathbb{C}^{n}$ and solves the union problem,
one of the most classical questions in Complex Analytic Geometry. When $X$ has dimension
$2$, we prove that the same result follows if we assume only that
$\Omega\subset\subset X$ is a domain of holomorphy in a Stein normal space. It is known,
however, that if $X$ is an arbitrary complex space which is exhaustable by an increasing
sequence of open Stein subsets $X_{1}\subset X_{2}\subset\dots\subset X_{n}\subset\dots$,
it does not follow in general that $X$ is holomorphically-convex
or holomorphically-separate (even if $X$ has no singularities). One can even obtain
$2$-dimensional complex manifolds on which all holomorphic functions are constant.