Is the intersection of a dense set and a linear subspace in $\mathbb R^n$ dense?

Question

Suppose $U \subset \mathbb R^n$ is open and open set $V$ is dense in $U$, i.e., $\bar{V} = U$. Let $\mathcal S$ be a linear subspace in $\mathbb R^n$. Putting $E = U \cap \mathcal S$, I am wondering whether $F := V \cap \mathcal S$ is dense in $E$ provided they are both nonempty.

Answer 1

No. With $U=\mathbb{R}^n$, $V=\{|x| < 1\} \cup \{x_n \neq 0\}$, $S=\{x_n=0\}$, $V \cap S$ is an non-empty bounded open subset of $S$.