Exercise 5, § 3

Let $X$ be a Hausdorff uniform space such that $X$ is the intersection of a countable family of open sets of $\hat{X}$. Show that if $Y$ is any Hausdorff uniform space which contains $X$ as a uniform subspace, then $X$ is the intersection of a closed subset of $Y$ and a countable family of open sets of $Y$.