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

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

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

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

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

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

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

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

When G is a compact topological group the following are equivalent:

Topological groups, together with their homomorphisms, form a category.

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

This holds in particular for all topological groups.

I. Structure of topological groups, integration theory, group representations.

This was to formulate class field theory for infinite extensions in terms of topological groups.

Topological groups began to be studied as such.

This page discusses a class of topological groups.

In mathematics, a class formation is a topological group acting on a module satisfying certain conditions.

In mathematics, the restricted product is a construction in the theory of topological groups.

This much is a fragment of a typical locally Euclidean topological group.

More generally, any topological group can be completed at a decreasing sequence of open subgroups.

As with topological groups, some authors require the topology to be Hausdorff.

Examples of non-abelian topological groups are given by the classical groups.

By homogeneity, local compactness for a topological group need only be checked at the identity.

Let 'G' be a locally compact topological group.

A basic example of a topological group with no small subgroup is the general linear group over the complex numbers.