---
tag: Y63847Z06N
kind: definition
name: Completion of a topological group
msc: [22A05, "54E15"]
defines: V1QDWHY4KT
uses: [7GKCGQ7215, P89SJ7JWJE, T3A3JTW6Q4, WZC4J62B1G]
sources:
  - url: https://en.wikipedia.org/wiki/Complete_topological_vector_space
---

## Definition

Let $G$ be a [[Hausdorff topological group|7GKCGQ7215]].
The *two-sided uniformity* of $G$ is the coarsest uniform structure finer than both the left and the right [[uniform structures|P89SJ7JWJE]] of $G$.
$G$ is *complete* if every [[Cauchy filter|WZC4J62B1G]] for the two-sided uniformity converges.
A *completion* of $G$ is a pair $(\hat G, \iota)$ such that:

1. $\hat G$ is a complete Hausdorff topological group;
2. $\iota \colon G \to \hat G$ is an [[isomorphism of topological groups|T3A3JTW6Q4]] from $G$ onto the subgroup $\iota(G)$ with the subspace topology;
3. $\iota(G)$ is dense in $\hat G$.

## Remarks

Every Hausdorff topological group has a completion, and any two completions are isomorphic by a unique isomorphism compatible with the maps $\iota$.
Example: $(\R, +)$ with the inclusion of $(\Q, +)$.
The group $\R$ is complete, $\Q$ carries the subspace topology, and $\Q$ is dense in $\R$.
Non-example: $(\R, +)$ with the inclusion of $(\Z, +)$ fails clause 3, since $\Z$ is closed in $\R$ and not dense.
For an abelian group the left, right and two-sided uniformities agree.
In general the completion for the left uniformity alone need not be a group.