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.
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2 Properties 3 Relationship to other areas of mathematics |
The real numbers R, together with addition as operation and its ordinary topology, form a topological group. More generally, Euclidean n-space R^{n} 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 R^{n×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 R^{3} 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.
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 T_{0} (i.e. Kolmogorov), then it is already T_{2} (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 T_{0} even if G is. It is therefore natural to restrict oneself to the category of T_{0} 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.
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.