Proposition 1

Let $X$ be a metric space and let $d$ be its metric. If $\hat{X}$ is the completion of $X$ (with respect to the uniformity defined by $d$), the function $d$ can be extended by continuity to $\hat{X} \times \hat{X}$; the extended function $\overline{d}$ is a metric on $\hat{X}$, and the uniformity of $\hat{X}$ coincides with that defined by the metric $\overline{d}$.

Proposition 1 of § 1, no. 3 shows that $\overline{d}$ is a finite pseudometric on $\hat{X}$, and that the uniformity defined by $\overline{d}$ on $\hat{X}$ is the uniformity obtained by completion; since this latter uniformity is Hausdorff, $\overline{d}$ is a metric. Whenever we consider the completion of a metric space $X$ as a metric space, it is always to be understood that the metric on $\hat{X}$ is that obtained by extending the metric on $X$ by continuity.