Let $X$ be a compact and Hausdorff space. Hence there is a uniformity $\mathcal{U}$ on $X$, also $X$ is normal space i.e. for every point $x\in X$ and every closed set $A\subseteq X$ there exist open sets $U, V$ such that $x\in U$, $A\subseteq V$ and $U\cap V=\emptyset$.
Can I say that there is an entourage $U\in \mathcal{U}$ such that $U[x]\cap U[A]=\emptyset$, where $U[x]=\{y: (x, y)\in U\}$?
Please help me to know it.
Start with an entourage $W$ such that $W[x]$ is disjoint from $A$. Since $\mathcal U$ is a uniformity, there is an entourage $U$ such that $U=U^{-1}$ and $U\circ U\subseteq W$. Then $U[x]\cap U[A]\varnothing$.