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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 $S\subseteq U[S]\subseteq W$?

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  • $\begingroup$ Cover $S$ by finitely many $U_i[s]$, $s \in S$ that all sit inside $W$. $\endgroup$ – Henno Brandsma Dec 24 '18 at 8:44
  • $\begingroup$ @HennoBrandsma, Yes it is true that there is finitely $U_i[s]$ that is cover for $S$ and all sit inside $W$, but I need one entorage $U\in \mathcal{U}$ with $S\subseteq U[S]\subseteq W$. $\endgroup$ – user479859 Dec 24 '18 at 9:52
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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 $$U[S]\subseteq U\left[\bigcup\{U_x[x]: x\in F\}\right] \subseteq \bigcup\{ U[U_x[x]]: x\in F\} \subseteq \bigcup\{ U^2_x[x]: x\in F\}]\subseteq W.$$

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