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Topological group

In mathematics, a topological group G is a group that is also a topological space such that the group multiplication
G × G -> G
and taking inverses
G -> G
are continuous maps. Here, G × G is viewed as a topological space by using the product topology. (See group object).

Though we do not do so here, it is common to also require that the topology on G be Hausdorff. The reasons, and some equivalent conditions, are discussed below.

Almost all objects investigated in analysis are topological groups (usually with some additional structure).

Every group can be made into a topological group by imposing the discrete topology on it. However, the more interesting situation is where the group has some other topology, not arising so directly from the group operation.

Examples

The real numbers R, together with addition as operation and its ordinary topology, form a topological group. More generally, Euclidean n-space Rn with addition and standard topology is a topological group. More generally still, all topological vector spaces, such as Banach spaces or Hilbert spaces, are topological groups.

The above examples are all abelian. Important examples of non-abelian topological groups are given by the Lie groups (topological groups that are also manifolds), for instance by the group GL(n,R) of all invertible n-by-n matrices with real entries. The topology on GL(n,R) is defined by viewing GL(n,R) as a subset of Euclidean space Rn×n.

All the examples above are Lie groups (if one views the infinite-dimensional vector spaces as infinite-dimensional "flat" Lie groups). An example of a topological group which is not a Lie group is given by the rational numbers Q. This is a countable space and it does not have the discrete topology. For a nonabelian example, consider the subgroup of rotations of R3 generated by two rotations by irrational multiples of 2π about different axes.

In every unitary Banach algebra, the set of invertible elements forms a topological group under multiplication.

Properties

If a is an element of a topological group G, then left or right multiplication with a yields a homeomorphism G -> G. This can be used to show that all topological groups are actually uniform spaces. Every topological group can be viewed as a uniform space in two ways; the left uniformity turns all left multiplications into uniformly continuous maps while the right uniformity turns all right multiplications into uniformly continuous maps. If G is not abelian, then these two need not coincide. The uniform structures allow to talk about notions such as completeness, uniform continuity and uniform convergence on topological groups.

As a uniform space, every topological group is completely regular. It follows that if a topological group is T0 (i.e. Kolmogorov), then it is already T2 (i.e. Hausdorff).

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

If H is a normal subgroup of the topological group G, then the factor group G/H becomes a topological group by using the quotient topology (the finest topology on G/H which makes the natural projection G -> G/H continuous), and the isomorphism theorems known from ordinary group theory remain valid in this setting. However, if H is not closed in the topology of G, then G/H won't be T0 even if G is. It is therefore natural to restrict oneself to the category of T0 topological groups, and restrict the definition of normal to normal and closed.

The algebraic and topological structures of a topological group interact in non-trivial ways. For example, in any topological group the connected component containing the identity element is a normal subgroup.

Relationship to other areas of mathematics

Of particular importance in harmonic analysis are the locally compact topological groups, because they admit a natural notion of measure and integral, given by the Haar measure. In many ways, the locally compact topological groups serve as a generalization of countable groups, while the compact topological groups can be seen as a generalization of finite groups. The theory of group representations is almost identical for finite groups and for compact topological groups.