# nLab topological abelian group

Topological abelian groups

# Topological abelian groups

## Definition

A topological abelian group (or TAG) is a topological group whose underlying group is abelian.

Topological abelian groups and continuous group homomorphisms form a category $Ab Top$.

## Remarks

Often TAGs are easier to understand than arbitrary topological groups, because no distinction need be made between left and right. For example, a TAG has only one uniform structure, a locally compact TAG has only one Haar measure (up to scale), etc.

## Applications

TAGs are important in analysis. For example, every topological vector space (TVS) is a TAG, and much of the theory of TVSs can be generalised to TAGs. See also G-norm.

## References

Last revised on July 13, 2010 at 16:44:40. See the history of this page for a list of all contributions to it.