Topological Groups

Topological Group

A topological group is a group that is also a topological space, such that the group operations of multiplication and inversion are both continuous maps. That is is a continuous mapping of the product space into .

Remark

In the language of category, topological groups can be defined concisely as group objects in the category of topological spaces .

e.g.

• Any group with discrete topology is a topological group;
• is a topological group. Moreover, any endomorphism of as a topological group is given by multiplication by a real number. However, as a group, has a discontinuous group endomorphism which is not given by multiplication by a real number (see counterexample, this counterexample requires axiom of choice to pick a Hamel basis). Consequently, an endomorphism of the topological group is given by for some integer ;
• Nonzero complex numbers with multiplication is a topological group, is a topological subgroup with multiplication;
• -Torus is a topological group with multiplication;
• General linear group is a topological group with matrix multiplication;
• Any subgroup of a topological group is a topological group with subspace topology;
• The (group) direct product of topological groups is a topological group with product topology.
  

Proposition

The left translation , on a topological group is a homeomorphism. Thus any topological group is topologically homogeneous.

Proof It is clearly invertible because . Continuity is straightforward from construction.

Proposition

Suppose is a topological group and is a subgroup. If is open, then it is also closed.

Proof Because the cosets of partition , and each coset is open (by left translation), the complement of is a union of open sets, hence open. Therefore, is closed.

Corollary

Let be a connected topological group and its identity element. If is any open neighborhood of , then is generated by . In other words, there are no subgroups of except for itself and the trivial subgroup .

Proof Let be an open neighborhood of . For each , let be the set of all products of at most elements of , and . Since each is open, is open. We now see that is also closed. For any , is an open neighbourhood of , so it must intersect . Let , then for some . Since , is a product of finitely many elements in , hence so is . Therefore, , and we have . Since is connected, we conclude that (ref. proposition).

Note that if is a topological group, and is a subgroup, then the coset space can be endowed with the quotient topology. Hence,

Proposition

Suppose is a topological group, and is a normal subgroup, then is also a topological group.

Topological Group Actions

Here we will give some examples of topological group actions.

e.g.

• Suppose is a subgroup of a topological group , then acts on by left translation, and the action is continuous and free;
• acts on by matrix multiplication. The only orbits are and . The induced quotient topology on the orbit space is , that is, the orbit space is homeomorphic to the Sierpiński space;
• Orthogonal group acts on by matrix multiplication. The only orbits are and the spheres of radius ;
• acts on by scalar multiplication, the quotient is the real projective space ;