Lemma 4

Let K be a compact topological space and let $\mathscr{R}$ be a set of open subsets of K covering K. There exists an entourage V of the uniform structure of K (TG, II, p. 27, th. 1) such that, for every $x\in K$, $V(x)$ is contained in one of the sets belonging to $\mathscr{R}$.

For every point $x$ of K, there exists an entourage $W_x$ of the uniform structure of K such that $W_x(x)$ is contained in one of the sets belonging to $\mathscr{R}$. Let $V_x$ be an entourage of the uniform structure of K such that $\overset{2}{V_{x}}$ is contained in $W_x$. The interiors of the $V_x(x)$ cover K; since the space K is compact, there exists a finite subset F of K such that the family $(V_y(y))_{y\in F}$ is a covering of K (TG, I, p. 59). Let V denote the intersection of the family $(V_y)_{y\in F}$; the set V is an entourage of the uniform structure of K. For every point $x\in K$, there exists a point $y\in F$ such that $x$ belongs to $V_y(y)$. Consequently, the set

$V(x)$ is contained in $\overset{2}{V_{y}}(y)$, hence in one of the sets belonging to $\mathscr{R}$.