Theorem 1

A Hausdorff topological group $G$ is isomorphic to a dense subgroup of a complete group $\hat{G}$ if and only if the image, under the symmetry $x \to x^{-1}$, of a Cauchy filter with respect to the right uniformity of $G$ is a Cauchy filter with respect to this uniformity. The complete group $\hat{G}$ (which is called the completion of $G$) is then unique (up to isomorphism).