Completion of a topological group

Summary

The completion of a Hausdorff topological group $G$ is a complete Hausdorff topological group that contains $G$ as a dense subgroup. Completeness refers to the uniform structure of the group, so it makes sense without a metric. Every Hausdorff topological group has a completion for its two-sided uniformity, unique up to isomorphism; this is the Raikov completion. For abelian groups this is the familiar completion with respect to Cauchy nets.