topological group


completely regularPlanetmathPlanetmathPlanetmath


A topological groupMathworldPlanetmath is a group G endowed with a topologyMathworldPlanetmath such that the multiplication and inverseMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath operationsMathworldPlanetmath of G are continuousPlanetmathPlanetmath ( That is, the map G×GG defined by (x,y)xy is continuous, where the topology on G×G is the product topology, and the map GG defined by xx-1 is also continuous.

Many authors require the topology on G to be HausdorffPlanetmathPlanetmath, which is equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath to requiring that the trivial subgroup be a closed setPlanetmathPlanetmath.

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


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 groupsMathworldPlanetmath are topological groups with additional structureMathworldPlanetmath.

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

Subgroups, quotients and products

Every subgroupMathworldPlanetmath ( of a topological group either has empty interior or is clopen. In particular, all proper subgroupsMathworldPlanetmath of a connected topological group have empty interior. The closureMathworldPlanetmathPlanetmath of any subgroup is also a subgroup, and the closure of a normal subgroupMathworldPlanetmath is normal (for proofs, see the entry “closure of sets closed under a finitary operation (”). 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 groupMathworldPlanetmath 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 (Gi)iI is a family of topological groups, then the unrestricted direct product iIGi is also a topological group, with the product topology.


Let G and H be topological groups, and let f:GH be a function.

The function f is said to be a homomorphism of topological groups if it is a group homomorphismMathworldPlanetmath 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 isomorphismMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath (that is, a bijectiveMathworldPlanetmath homomorphism of topological groups) and yet not be an isomorphism of topological groups. This occurs, for example, if G is with the discrete topology, and H is with its usual topology, and f is the identity map on .

Topological properties

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

Every topological group is bihomogeneous and completely regular ( 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 T1 (, and being completely regular is therefore Hausdorff. A topological group is not necessarily, however, a counterexample being the unrestricted direct product of uncountably many copies of the discrete group .

Every topological group is obviously an H-spaceMathworldPlanetmath. Consequently, the fundamental groupMathworldPlanetmathPlanetmath of a topological group is abelianMathworldPlanetmath. Note that because topological groups are homogeneousPlanetmathPlanetmathPlanetmath, the fundamental group does not depend (up to isomorphism) on the choice of basepoint.

Every locally compact topological group is and strongly paracompact.

Every connected locally compact topological group is σ-compactPlanetmathPlanetmath.

Other notes

Every topological group possesses a natural uniformity, which induces the topology. See the entry about the uniformity of a topological group (

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