Exercise 1, § 4

Let $\mathfrak{H}$ be an open covering of a compact space $X$. Show that there is an entourage $V$ of the uniformity of $X$ such that, for each $x \in X$, $V(x)$ is contained in a set belonging to $\mathfrak{H}$ [remark that for each $x \in X$ there is an entourage $W_x$ such that $\dot{W}_x(x)$ is contained in a set of $\mathfrak{H}$, and cover $X$ by a finite number of sets $W_x(x)$].