---
tag: V1QDWHY4KT
kind: concept
name: Completion of a topological group
msc: [22A05, "54E15"]
sources:
  - url: https://en.wikipedia.org/wiki/Complete_topological_vector_space
---

## Summary

The completion of a [[Hausdorff topological group|7GKCGQ7215]] $G$ is a complete Hausdorff topological group that contains $G$ as a dense subgroup.
Completeness refers to the [[uniform structure|P89SJ7JWJE]] 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.

## Explanations

### L1 human

The rational numbers have holes: a sequence of rationals can get closer and closer together without having a rational limit.
Filling in all such holes in the additive group of rationals gives the real numbers.
The completion of a topological group does the same thing in general: it adds limits for all the Cauchy nets, and the group law extends to the new points.
For example, the completion of the integers with the $p$-adic topology is the group of $p$-adic integers.