Let X be a Stein manifold and U an open, connected, relatively compact, holomorphically convex subset of X. Is the closure of U in X holomorphically convex?


Also, if X is a Stein *space* with a finite number of singularities, and U is an open, connected, relatively compact, holomorphically convex subset of X containing all the singularities, is it true that the closure of U is holomorphically convex in this case too?



Is the closure of an open holomorphically convex subset of a Stein space holomorphically convex?
The answer is **no**.


In fact, in the first page of the paper


P.J. de Paepe, "Closures of proper analytic polyedra", Compositio mathematica 28 (1974), p. 333-341


there is the example of a relatively compact Stein space $U \subset \mathbb{C}^2$ such that its closure $\bar{U}$ is **not** holomorphically convex.


The example is as follows:


$U:=\{(z, w) \in \mathbb{C}^2 \, | \, |z| < |w|, \frac{1}{2} < |w| <1 \} \cup \{(z, w) \in \mathbb{C}^2 \, | \, |w| < |z|, \frac{1}{2} < |z| <1 \} $.


In fact, the smallest holomorphic set containing $\bar{U}$ is the unit polidisk in $\mathbb{C}^2$, which is different from $\bar{U}$.


**EDIT**. In a previous version of this post I wrote that the answer to the question was positive, since I erroneously assumed that the closure of $U$ was a complex space, see BCnrd's comment below.


