# Connectedness of boundary of a Stein domain

Let $$Y$$ be a Stein manifold and $$D\subset\subset Y$$ be a Stein domain. I think $$\overline D$$ has connected boundary, and it should be somewhere, but I cannot find a reference for this. Thanks

• You need to assume Y has complex dimension at least two. Oct 4, 2021 at 15:59

• Thank you very much. In case $Y$ has dimension 1, it is a connected Riemann Surface, but this doesn't imply that is has connected boundary. Can't we say anything in this case? Just about $Y$, no matter of subsets. Thanks