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

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

• Cover $S$ by finitely many $U_i[s]$, $s \in S$ that all sit inside $W$. – Henno Brandsma Dec 24 '18 at 8:44
• @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 $S\subseteq U[S]\subseteq W$. – user479859 Dec 24 '18 at 9:52

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 $$U[S]\subseteq U\left[\bigcup\{U_x[x]: x\in F\}\right] \subseteq \bigcup\{ U[U_x[x]]: x\in F\} \subseteq \bigcup\{ U^2_x[x]: x\in F\}]\subseteq W.$$