gn.general topology - A question about disconnecting a Euclidean space or a Hilbert space

Assume the complement of $S$ in $mathbb{R}^n$ is not connected, say $A$ and $B$ are relatively closed and disjoint in $mathbb{R}^nsetminus S$ (and nonempty of course); let $O$ be the complement of the closure of $B$ and $U$ the complement of the closure of $A$, then $O$ and $U$ are disjoint nonempty open subsets of $mathbb{R}^n$ and the complement of their union, $F$, is closed in $mathbb{R}^n$, a subset of $S$ and it separates $mathbb{R}^n$.
In short: $S$ contains a closed set that also separates; as you noted that set is zero-dimensional and hence the answer is `no' for Euclidean spaces.
I don't know (yet) about Hilbert space.