Proposition 5.1.18. Let $(X, \fU)$ be a uniform space and $A \subset X$ be a dense subset, then $\bracsn{\overline{U}: U \in \fU_A}$ forms a fundamental system of entourages for $X$.
Proof. Let $U \in \fU$ be an open entourage, then by (3) of Definition 4.5.5, $\overline{U \cap (A \times A)}= \overline{U}$ for all $U \in \fU$, so $\overline{U \cap (A \times A)}$ is an entourage. By Proposition 5.1.14, every closed entourage of $X$ contains an element of $\bracsn{\overline{U}: U \in \fU_A}$.$\square$