More generally, every commutative topological group is also a uniform space.

This need not be true in a general abelian topological group (see examples below).

A generalization to functions g taking values in any topological group is also possible.

With respect to this topology, G is a topological group.

It is known, however, that local topological groups do not necessarily have global counterparts.

The identity component of a topological group is always a characteristic subgroup.

The real numbers R, together with addition as operation and its usual topology, form a topological group.

As a uniform space, every topological group is completely regular.

For example, a homomorphism of topological groups is often required to be continuous.

Thus SO(n) is a classic example of a topological group.