Proposition 14

Let X be a complete Hausdorff uniform space, $\mathfrak{U}$ its uniformity, and let Z be a dense subspace of X. If $\mathfrak{U}'$ is a uniformity on X which is coarser than $\mathfrak{U}$ and which induces the same uniformity as $\mathfrak{U}$ on Z, then $\mathfrak{U} = \mathfrak{U}'$.

Let X' denote the set X with the uniformity $\mathfrak{U}'$. The composition of the canonical mapping $X' \to \hat{X}'$ and the identity mapping $X \to X'$ is a uniformly continuous mapping $\varphi : X \to \hat{X}'$. Since Z is Hausdorff for the uniform structure induced by $\mathfrak{U}'$, the restriction of $\varphi$ to Z is by hypothesis an isomorphism of Z onto the dense subspace $\varphi(Z)$ of $\hat{X}'$; it follows (no. 6, Corollary to Theorem 2) that $\varphi$ itself is an isomorphism of X onto $\hat{X}'$, hence $X' = \hat{X}'$ and $\mathfrak{U}' = \mathfrak{U}$.