Theorem 2

Let $G$ be a commutative topological group. Then the functions $x^{-1}$ and $xy$ are uniformly continuous on $G$ and $G \times G$ respectively. Moreover $G$ admits a Hausdorff completion $\hat{G}$, and $\hat{G}$ is commutative.

The uniform continuity of $x^{-1}$ follows from Proposition 2 of no. 1, and that of $xy$ from Proposition 3 of no. 1, since $(x, y) \to xy$ is a continuous homomorphism of $G \times G$ into $G$. If $G$ is Hausdorff, it satisfies the condition of Theorem 1 of no. 4 (as does every Hausdorff group whose left and right uniformities coincide); moreover the functions $xy$ and $yx$ are equal on $\hat{G} \times \hat{G}$ by the principle of extension of identities; hence the second part of the theorem, by considering in the general case the Hausdorff group associated with $G$.

It follows in particular from this theorem that if $f$ and $g$ are two uniformly continuous mappings of a uniform space $X$ into a commutative group $G$, written additively, then the functions $-f$ and $f + g$ are uniformly continuous.