Let Y be an open subset of a reduced compact complex space X such that X - Y is the support of an effective divisor D. If X is a surface and D is an effective Weil divisor, we give sufficient conditions so that Y is Stein. If X is of pure dimension d ≥ 1 and X - Y is the support of an effective Cartier divisor D, we show that Y is Stein if Y contains no compact curves, $H^i\left(Y{,}_Y\right)=0$ for all i > 0, and for every point x₀ ∈ X-Y there is an n ∈ ℕ such that ${\Phi }_{\left|nD\right|}^{-1}\left({\Phi }_{\left|nD\right|}\left(x₀\right)\right)\cap Y$ is empty or has dimension 0, where ${\Phi }_{\left|nD\right|}$ is the map from...