# topological group

## Definitions

A topological group  is a group $G$ endowed with a topology  such that the multiplication and inverse      operations  of $G$ are continuous  (http://planetmath.org/Continuous). That is, the map $G\times G\to G$ defined by $(x,y)\mapsto xy$ is continuous, where the topology on $G\times G$ is the product topology, and the map $G\to G$ defined by $x\mapsto x^{-1}$ is also continuous.

A topology on a group $G$ that makes $G$ into a topology group is called a group topology for $G$.

## Examples

Any group becomes a topological group if it is given the discrete topology.

Any group becomes a topological group if it is given the indiscrete topology.

The real numbers with the standard topology form a topological group. More generally, an ordered group with its is a topological group.

Profinite groups are another important class of topological groups; they arise, for example, in infinite Galois theory.

## Subgroups, quotients and products

If $G$ is a topological group and $N$ is a normal subgroup of $G$, then the quotient group  $G/N$ is also a topological group, with the quotient topology. This quotient $G/N$ is Hausdorff if and only if $N$ is a closed subset of $G$.

If $(G_{i})_{i\in I}$ is a family of topological groups, then the unrestricted direct product $\prod_{i\in I}G_{i}$ is also a topological group, with the product topology.

## Morphisms

Let $G$ and $H$ be topological groups, and let $f\colon G\to H$ be a function.

The function $f$ is said to be a homomorphism of topological groups if it is a group homomorphism  and is also continuous. It is said to be an isomorphism of topological groups if it is both a group isomorphism and a homeomorphism.

## Topological properties

Every topological group is bihomogeneous and completely regular (http://planetmath.org/Tychonoff). Note that our earlier claim that a topological group is Hausdorff if and only if its trivial subgroup is closed follows from this: if the trivial subgroup is closed, then homogeneity ensures that all singletons are closed, and so the space is $\hbox{T}_{1}$ (http://planetmath.org/T1Space), and being completely regular is therefore Hausdorff. A topological group is not necessarily http://planetmath.org/node/1530normal, however, a counterexample being the unrestricted direct product of uncountably many copies of the discrete group $\mathbb{Z}$.

Every locally compact topological group is http://planetmath.org/node/1530normal and strongly paracompact.

## Other notes

Every topological group possesses a natural uniformity, which induces the topology. See the entry about the uniformity of a topological group (http://planetmath.org/UniformStructureOfATopologicalGroup).

A locally compact topological group possesses a natural measure, called the Haar measure.

Title topological group TopologicalGroup 2013-03-22 15:47:09 2013-03-22 15:47:09 yark (2760) yark (2760) 29 yark (2760) Definition msc 22A05 Group TopologicalSpace homomorphism of topological groups isomorphism of topological groups