For every closed neighborhood $\Delta_X\subset D$ , Is there an entourage $U$ with $U\subseteq D$?

Question

Let $(X, \mathcal{U})$ be an uniform space. It is known that every entourage $U\in\mathcal{U}$ is a neighborhood of $\Delta_X$, but the converse is not true, in general.

What can say about closed neighborhood of $\Delta_X$? Is it true that for a closed neighborhood $D\neq \Delta_X$ of $\Delta_X$, there is $U\in\mathcal{U}$ with $U\subseteq D$?

Thanks a lot.

Answer 1

No, that's not true in general. Consider $\mathbb{R}$ with its standard uniform structure. Let$$D^\star=\left\{(x,y)\in\mathbb{R}^2\,\middle|\,-\frac1{1+x^2}\leqslant y\leqslant\frac1{1+x^2}\right\}$$and let $D$ be what you obtain when you apply to $D^\star$ a rotation of $\frac\pi4$ radians around the origin. Then $D$ is a closed neighborhood of $\Delta_{\mathbb R}$, but it contains no entourage.

Answer 2

Show that in a separated uniform space: If we have a neighbourhood $U$ of $\Delta_X$ that is no entourage, then $\overline{U}$ is a closed neighbourhood of $\Delta_X$ that contains no entourage.