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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?

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  • $\begingroup$ Dea @user3566, what is your definition of "holomorphically convex"? If you mean $H(X)$-convexity defined in p.472 of Rossi's paper you cited, then an annulus in $\mathbb{C}$ possesses a fundamental system of neighbourhoods which are Stein opens, while $K$ is not $H(\mathbb{C})$-convex, see p.477 loc.cit. $\endgroup$
    – Doug Liu
    Commented Mar 3, 2023 at 16:32

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The answer to this question is yes, $K$ is necessarily holomorphically convex. See p.161 of "Topological Algebras Selected Topics" by A. Mallios,

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