For closed set $S$ in uniform space $(X, \mathcal{U})$, with $S\subseteq W$, is there $U\in \mathcal{U}$ such that $S\subseteq U[S]\subseteq W$?

Question

Let $(X, \mathcal{U})$ be a compact, Hausdorff uniform space and $S\subseteq X$ be a closed set with $S\subseteq W$, where $W\subseteq X$ is an open set in $X$.

Let $U[x]=\{y: (x, y)\in U\}$ and $U[S]=\cup_{x\in S}U[x]$.

Is there $U\in \mathcal{U}$ such that $S\subseteq U[S]\subseteq W$?

Answer 1

Is there $U\in \mathcal{U}$ such that $S\subseteq U[S]\subseteq W$?

Yes. For each $x\in S$ pick a symmetric entourage $U_x\in\mathcal U$ such that $U_x^2[x]\subseteq W$. Since the set $S$ is compact, there exists a finite subset $F$ of $S$ such that $S\subseteq\bigcup\{U_x[x]: x\in F\} $. Put $U=\bigcap \{U_x:x\in F\}$. Then $$U[S]\subseteq U\left[\bigcup\{U_x[x]: x\in F\}\right] \subseteq \bigcup\{ U[U_x[x]]: x\in F\} \subseteq \bigcup\{ U^2_x[x]: x\in F\}]\subseteq W.$$