TITLE: Is $\overline{D}\subseteq D\circ D$ in a uniform space?
QUESTION [5 upvotes]: Suppose $(X,\mathcal{D})$ is a uniform space and $D\in\mathcal{D}$. Is it true that 
$$\overline{D}\subseteq D\circ D,$$?
here we use the product topology to define $\overline{D}$.

REPLY [2 votes]: I'm not sure but I think this is a counterexample:
Let $\mathcal D$ be the normal uniformity on $\Bbb R$ made by the Euclidean  metric $d$. Let
$$D=U_d(1)\cup \Bbb Q^2$$
where
$$U_d(1)=\{(x,y)\in \Bbb R^2 \mid d(x,y)\le 1 \}$$
Then
$$(-\sqrt 2,\sqrt 2)\in \overline D$$
But
$$(-\sqrt 2,\sqrt 2)\notin D\circ D$$