For entourage $U_0\in\mathcal{U}$, is there a closed entourage $U_1\subseteq U_0$?

Question

Let $(X, \mathcal{U})$ be an uniform space. I know that for closed neighborhood $D$ of $\Delta_X$, there is no an entourage $U\in \mathcal{U}$ with $U\subseteq D$, in general.

Is it true for an entourage $U_0\in\mathcal{U}$, there is a closed entourage $U_1\in\mathcal{U}$ with $U_1\subseteq U_0$?

Please help me to know it.

Answer 1

It is well-known that in a uniform space, that if $V^3 \subseteq U$ then $\overline{V} \subseteq U$.