Definitions

A topological group is a group endowed with a topology such that the multiplication and inverse operations of are 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 to be Hausdorff, which is equivalent to requiring that the trivial subgroup be a closed set.

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

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 order topology 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 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”). A subgroup of a topological group is itself a topological group, with the subspace topology.

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

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.

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. 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 $\hbox{T}_1$ , and being completely regular is therefore Hausdorff. A topological group is not necessarily normal, 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 normal and strongly paracompact.

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