# 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.

Many authors require the topology on $G$ to be Hausdorff^{},
which is equivalent^{} to requiring that the trivial subgroup be a closed set^{}.

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.

Lie groups^{} are topological groups with additional structure^{}.

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

## Subgroups, quotients and products

Every subgroup^{} (http://planetmath.org/Subgroup) of a topological group
either has empty interior or is clopen.
In particular, all proper subgroups^{} of a connected topological group
have empty interior.
The closure^{} of any subgroup is also a subgroup,
and the closure of a normal subgroup^{} is normal
(for proofs, see the entry
“closure of sets closed under a finitary operation (http://planetmath.org/ClosureOfSetsClosedUnderAFinitaryOperation)”).
A subgroup of a topological group is itself a topological group,
with the subspace topology.

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 $f:G\to H$ 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.

Note that it is possible for $f$ to be a continuous group isomorphism^{}
(that is, a bijective^{} homomorphism of topological groups)
and yet not be an isomorphism of topological groups.
This occurs, for example, if $G$ is $\mathbb{R}$ with the discrete topology,
and $H$ is $\mathbb{R}$ with its usual topology,
and $f$ is the identity map on $\mathbb{R}$.

## Topological properties

While every group can be made into a topological group,
the same cannot be said of every topological space.
In this section^{} we mention some of the properties
that the underlying topological space must have.

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 ${\text{T}}_{1}$ (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 topological group is obviously an H-space^{}.
Consequently, the fundamental group^{} of a topological group is abelian^{}.
Note that because topological groups are homogeneous^{},
the fundamental group does not depend (up to isomorphism)
on the choice of basepoint.

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

Every connected locally compact topological group is $\sigma $-compact^{}.

## 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 |
---|---|

Canonical name | TopologicalGroup |

Date of creation | 2013-03-22 15:47:09 |

Last modified on | 2013-03-22 15:47:09 |

Owner | yark (2760) |

Last modified by | yark (2760) |

Numerical id | 29 |

Author | yark (2760) |

Entry type | Definition |

Classification | msc 22A05 |

Related topic | Group |

Related topic | TopologicalSpace |

Defines | homomorphism of topological groups |

Defines | isomorphism of topological groups |