Proposition 6

Let $A$ be a dense subset of a uniform space $X$. Then the closures, in $X \times X$, of the entourages of the uniform subspace $A$ form a fundamental system of entourages of $X$.

$A \times A$ is dense in $X \times X$ (Chapter I, § 4, no. 3, Proposition 7). Let $V$ be an open entourage of $A$; it is the intersection of $A \times A$ with an open entourage $U$ of $X$. We have $U \subset \overline{V}$ (Chapter I, § 1, no. 6, Proposition 5), and this relation together with $\overline{V} \subset \overline{U}$ establishes the proposition, in view of § 1, no. 2, Corollary 2 of Proposition 2.