Proposition 3

The extended real line is compact.

Hence (Chapter II, § 4, no. 1, Theorem 1) there is a unique uniformity on $\overline{\mathbf{R}}$ compatible with its topology; this uniformity is isomorphic with the uniformity induced on $[a, b]$ by the additive uniformity of $\mathbf{R}$. But it should be remarked that the uniformity induced on $\mathbf{R}$ by that of $\overline{\mathbf{R}}$ is not the additive uniformity of $\mathbf{R}$ (although it is compatible with the topology of the real line); for $\mathbf{R}$ is a complete space with respect to its additive uniformity, but is not a complete subspace of $\overline{\mathbf{R}}$, since it is not closed in $\overline{\mathbf{R}}$.