In mathematics, a topological group G is a group that is also a topological space such that the group multiplication G × G → G and the inverse operation G → G are continuous maps. Here, G × G is viewed as a topological space by using the product topology. Although we do not do so here, it is common to also require that the topology on G be Hausdorff, namely that any two points in this space have disjoint neighborhoods. The reasons, and some equivalent conditions, are discussed below. In the language of category theory, one would say that topological groups are group objects in the category of topological spaces.
Almost all objects investigated in analysis are topological groups (usually with some additional structure).
Every group can be trivially made into a topological group by considering it with the discrete topology; such groups are called discrete groups. In this sense, the theory of topological groups subsumes that of ordinary groups.
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, the additive groups of all topological vector spaces, such as Banach spaces or Hilbert spaces, are topological groups.
The above examples are all abelian. Examples of non-abelian topological groups are given by Lie groups (topological groups that are also manifolds). For instance, the general linear group GL(n,R) of all invertible n-by-n matrices with real entries can be viewed as a topological group with the topology defined by viewing GL(n,R) as a subset of Euclidean space Rn×n. All Lie groups are locally compact.
An example of a topological group which is not a Lie group is given by the rational numbers Q with the topology inherited from R. 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 Banach algebra with multiplicative identity, the set of invertible elements forms a topological group under multiplication.
The algebraic and topological structures of a topological group interact in non-trivial ways. For example, in any topological group the identity component (i.e. the connected component containing the identity element) is a closed normal subgroup.
The inversion operation on a topological group G gives a homeomorphism from G to itself. Likewise, if a is any element of G, then left or right multiplication by a yields a homeomorphism G → G.
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.
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.
Every subgroup of a topological group is itself a topological group when given the subspace topology. If H is a subgroup of G the set of left or right cosets G/H is a topological space when given the quotient topology (the finest topology on G/H which makes the natural projection q : G → G/H continuous). One can show that the quotient map q : G → G/H is always open.
If H is a normal subgroup of G, then the factor group, G/H becomes a topological group, 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.
If H is a subgroup of G then the closure of H is also a subgroup. Likewise, if H is a normal subgroup, the closure of H is normal.
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.