Increasing unions of Stein spaces with singularities

We show that if X is a Stein space and, if Ω⊂X is exhaustable by a sequence Ω1⊂Ω2⊂…⊂Ωn⊂… of open Stein subsets of X, then Ω is Stein. This generalizes a well-known result of Behnke and Stein which is obtained for X=Cn 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 Ω⊂⊂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 X1⊂X2⊂⋯⊂Xn⊂…, 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.