Topological group

In mathematics, a topological group is a group that has a group structure with the " acceptable " topology. The topological structure allows for example to consider limits in this group, and to speak of continuous homomorphisms.

Definition

A group G is called topological group, if it is provided with a topology such that:

Examples

The real numbers R with the addition and the usual topology form a topological group. More generally, the n-dimensional Euclidean space Rn with the vector addition and the standard topology is a topological group. Also, every Banach space and Hilbert space is a topological group with respect to addition.

The above examples are all abelian. An important example of a non-Abelian topological group is the group GL ( n, R) of all invertible real n-by- n matrices. The topology results in this by one perceives this group as a subset of the Euclidean vector space Rn × n.

Rn is as GL (n, R) Lie group, i.e., a topological group, in which the topological structure is that of a manifold.

An example of a topological group which is not a Lie group, the additive group of rational numbers Q is (she is a countable set, which is not provided with the discrete topology). A non- abelian example is the subgroup of the rotation group of R3, which is generated by two rotations by irrational multiples of π (the circle Pi ) about different axes.

In each unitary Banach algebra the set of invertible elements with multiplication forms a topological group.

Properties

The algebraic and topological structure of a topological group G are closely linked. Thus, for example, in an arbitrary topology connected component group of the neutral element, a normal completed subassembly by G.

A is an element of a topological group G, then the left and right multiplication a multiplication homeomorphisms from G to G as well as the inverse mapping.

Each topological group can be regarded as a uniform space. Two elementary uniform structures that arise from the group structure, are the left and right uniform structure. The left uniform structure makes the left multiplication is uniformly continuous, the right uniform structure, makes the right multiplication is uniformly continuous. For non- Abelian groups, these two uniform structures differ in general. The uniform structures allow particular terms such as completeness, uniform continuity and uniform convergence defined.

Like any topology generated by a uniform space, the topology of a topological group is completely regular. In particular, that a topological group which satisfies T0 (i.e., if it is a Kolmogorov space ), even is a Hausdorff space.

The most natural notion of homomorphism between topological groups is that of a continuous group homomorphism. The topological groups together with the continuous group homomorphisms form a category.

Every subgroup of a topological group with the subspace topology again a topological group. For a subgroup H of G, the left and right cosets G / H form a topological space with the quotient topology.

If H is a normal subgroup of G, then G / H is a topological group. Note, however, that if H is in the topology of G is not complete, the resulting topology on G / H does not satisfy the axiom T0. It is therefore natural to restrict to the category of T0 topological groups and to define the notion of normal as normal and complete.

If H is a subgroup of G, then the closed hull of H is again a subgroup. Similarly, the conclusion of a normal subgroup is normal.