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

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.

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.
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.