Exercise 7, § 1

Let $A$ be a non-empty connected open set in $\mathbf{R}^n$ ($n > 1$) and let $(V_p)_{p \in \mathbf{N}}$ be a countable family of linear varieties in $\mathbf{R}^n$, each of which is of dimension $\leq n - 2$. Show that if $B$ is the union of the $V_p$, then $A \cap \overline{B}$ is dense in $A$ and connected. (To show that $A \cap \overline{B}$ is dense in $A$, use induction on $n$. Then show that if $x, y$ are two distinct points of $A$, there exists for each $\varepsilon > 0$ a point $y' \in A$ such that $\|y' - y\| \leq \varepsilon$ and such that the closed segment with end-points $x$ and $y'$ meets $B$ only at $x$. Finally use Exercise 6 to show that $A \cap \overline{B}$ is connected.)