Exercise 10, § 4

Let X be a Hausdorff uniform space which has an entourage V such that V(x) is precompact for each x ∈ X. Show that the completion $\hat{X}$ is locally compact and satisfies the conditions of Exercise 9. [If W is a symmetric open entourage of X such that $\hat{W} \subset V$, show that the closure $\overline{W}$ of W in $\hat{X} \times \hat{X}$ is such that $\overline{W}(x)$ is compact for each x ∈ $\hat{X}$].