On a Theorem of Lelong

The following two results are proved: (1) Let X be an open subspace of the Stein space S of bounded dimension. If for every holomorphic function f which is nonconstant on each irreducible component of positive dimension of S, one has (Y = {f = 0} cap X) is Stein, and if the restriction map of holomorphic maps to Y has dense image, then X is Stein. (2) Let G be an unramified Riemann domain over a Stein manifold M of dimension n, and let ( ˆ G) be its extension to an unramified Riemann domain over C^(2n+1). Let k be a fixed integer, 2 ≤ k ≤ 2n. Then G is Stein if and only if the preimage (ˆ H) is Stein for every complex k-plane H of C^(2n+1).