Proposition 1

Let $X$ be a Hausdorff uniform space whose uniformity $\mathscr{U}$ is defined by a family of finite pseudometrics $(f_i)$, and let $\hat{X}$ be the completion of $X$. Then the functions $f_i$ can be extended by continuity to $\hat{X} \times \hat{X}$; the extended functions $\bar{f}_i$ are finite pseudometrics on $\hat{X} \times \hat{X}$, and the family $(f_i)$ defines the uniformity of $\hat{X}$.

First, the $f_i$ can be extended by continuity to $\hat{X} \times \hat{X}$, because they are uniformly continuous on $X \times X$; and the extended functions $\bar{f}_i$ are uniformly continuous on $\hat{X} \times \hat{X}$ (Chapter II, § 3, no. 6, Theorem 2); moreover, they are pseudometrics on $\hat{X}$ by virtue of the principle of extension of inequalities (Chapter IV, § 5, no. 2, Theorem 1). Let $\mathscr{U}_1$ denote the uniformity on $\hat{X}$ obtained by completion, and let $\mathscr{U}_2$ denote the uniformity defined by the family of pseudometrics $(\bar{f}_i)$. Then $\mathscr{U}_2$ is coarser than $\mathscr{U}_1$; for each $\bar{f}_i$ is uniformly continuous on $\hat{X} \times \hat{X}$ with respect to $\mathscr{U}_1$, and hence for each $a > 0$ there exists an entourage $V$ of $\mathscr{U}_1$ such that, whenever $(x, y) \in V$, we have $|\bar{f}_i(x, y) - \bar{f}_i(x, x)| \leq a$, that is [since $\bar{f}_i(x, x) = 0$], $V \subset \bar{f}_i([0, a])$; hence every entourage of $\mathscr{U}_2$ is an entourage of $\mathscr{U}_1$. On the other hand, $\mathscr{U}_1$ and $\mathscr{U}_2$ induce the same uniformity $\mathscr{U}$ on $X$. As $\hat{X}$ is complete with respect to $\mathscr{U}_1$, it follows that $\mathscr{U}_1$ and $\mathscr{U}_2$ are identical (Chapter II, § 3, no. 7, Proposition 14).