Let X be a Stein manifold and let K be a compact subset of X. Suppose that K possesses in X a fundamental system of neighbourhoods which are Stein spaces. Then, it is a result by Rossi that such a compact subset K is holomorphically convex (see [Rossi, "Holomorphically convex sets in several complex variables," Ann. Math., 74, No. 3, 470-493 (1961)]).


Now let X be a Stein *space* with a finite number of singularities and let K be a connected compact subset of X containing all the singularities. Suppose that K possesses in X a fundamental system of neighbourhoods which are Stein spaces. Is it then necessarily true that K is holomorphically convex?



Is a compact subset of a Stein space admitting a fundamental system of Stein neighbourhoods necessarily holomorphically convex?
The answer to this question is yes, $K$ is necessarily holomorphically convex. See p.161 of "Topological Algebras Selected Topics" by A. Mallios, 


