Exercise 3, § 1

Let $V$ be that entourage of the additive uniformity on $\mathbf{R}$ which consists of all pairs $(x, y)$ such that either $|x - y| \leq 1$ or $xy \geq 1$. Show that $V$ is closed in $\mathbf{R} \times \mathbf{R}$, but that if $A$ denotes the (closed) subset of $\mathbf{R}$ consisting of all integers $n \geq 2$, then $V(A)$ is not closed in $\mathbf{R}$*