Topological group

In mathematics, topological groups are the combination of groups and topological spaces, i.e. they are groups and topological spaces at the same time, such that the continuity condition for the group operations connects these two structures together and consequently they are not independent from each other.
Topological groups were studied extensively in the period of 1925 to 1940. Haar and Weil showed that the integrals and Fourier series are special cases of a construct that can be defined on a very wide class of topological groups.
Topological groups, along with continuous group actions, are used to study continuous symmetries, which have many applications, for example, in physics. In functional analysis, every topological vector space is an additive topological group with the additional property that scalar multiplication is continuous; consequently, many results from the theory of topological groups can be applied to functional analysis.

Formal definition

A topological group,, is a topological space that is also a group such that the group operation :
and the inversion map:
are continuous.
Here is viewed as a topological space with the product topology.
Such a topology is said to be compatible with the group operations and is called a group topology.
;Checking continuity
The product map is continuous if and only if for any and any neighborhood of in, there exist neighborhoods of and of in such that, where.
The inversion map is continuous if and only if for any and any neighborhood of in, there exists a neighborhood of in such that where
To show that a topology is compatible with the group operations, it suffices to check that the map
is continuous.
Explicitly, this means that for any and any neighborhood in of, there exist neighborhoods of and of in such that.
;Additive notation
This definition used notation for multiplicative groups;
the equivalent for additive groups would be that the following two operations are continuous:
;Hausdorffness
Although not part of this definition, many authors require that the topology on be Hausdorff.
One reason for this is that any topological group can be canonically associated with a Hausdorff topological group by taking an appropriate canonical quotient;
this however, often still requires working with the original non-Hausdorff topological group.
Other reasons, and some equivalent conditions, are discussed below.
This article will not assume that topological groups are necessarily Hausdorff.
;Category
In the language of category theory, topological groups can be defined concisely as group objects in the category of topological spaces, in the same way that ordinary groups are group objects in the category of sets.
Note that the axioms are given in terms of the maps, hence are categorical definitions.

Homomorphisms

A homomorphism of topological groups means a continuous group homomorphism.
Topological groups, together with their homomorphisms, form a category.
A group homomorphism between topological groups is continuous if and only if it is continuous at some point.
An isomorphism of topological groups is a group isomorphism that is also a homeomorphism of the underlying topological spaces.
This is stronger than simply requiring a continuous group isomorphism—the inverse must also be continuous.
There are examples of topological groups that are isomorphic as ordinary groups but not as topological groups.
Indeed, any non-discrete topological group is also a topological group when considered with the discrete topology.
The underlying groups are the same, but as topological groups there is not an isomorphism.

Examples

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 indiscrete topology also makes every group into a topological group.
The real numbers, with the usual topology form a topological group under addition.
Euclidean -space is also a topological group under addition, and more generally, every topological vector space forms an topological group.
Some other examples of abelian topological groups are the circle group, or the torus for any natural number.
The classical groups are important examples of non-abelian topological groups. For instance, the general linear group of all invertible -by- matrices with real entries can be viewed as a topological group with the topology defined by viewing as a subspace of Euclidean space.
Another classical group is the orthogonal group, the group of all linear maps from to itself that preserve the length of all vectors.
The orthogonal group is compact as a topological space. Much of Euclidean geometry can be viewed as studying the structure of the orthogonal group, or the closely related group of isometries of.
The groups mentioned so far are all Lie groups, meaning that they are smooth manifolds in such a way that the group operations are smooth, not just continuous.
Lie groups are the best-understood topological groups; many questions about Lie groups can be converted to purely algebraic questions about Lie algebras and then solved.
An example of a topological group that is not a Lie group is the additive group of rational numbers, with the topology inherited from.
This is a countable space, and it does not have the discrete topology.
An important example for number theory is the group of p-adic integers, for a prime number, meaning the inverse limit of the finite groups as n goes to infinity.
The group is well behaved in that it is compact, but it differs from Lie groups in that it is totally disconnected.
More generally, there is a theory of p-adic Lie groups, including compact groups such as as well as locally compact groups such as , where the locally compact field of p-adic numbers.
The group is a profinite group; it is isomorphic to a subgroup of the product in such a way that its topology is induced by the product topology, where the finite groups are given the discrete topology.
Another large class of profinite groups important in number theory are absolute Galois groups.
Some topological groups can be viewed as infinite dimensional Lie groups; this phrase is best understood informally, to include several different families of examples.
For example, a topological vector space, such as a Banach space or Hilbert space, is an abelian topological group under addition. Some other infinite-dimensional groups that have been studied, with varying degrees of success, are loop groups, Kac–Moody groups, Diffeomorphism groups, homeomorphism groups, and gauge groups.
In every Banach algebra with multiplicative identity, the set of invertible elements forms a topological group under multiplication.
For example, the group of invertible bounded operators on a Hilbert space arises this way.

Properties

Translation invariance

Every topological group's topology is, which by definition means that if for any left or right multiplication by this element yields a homeomorphism
Consequently, for any and the subset is open in if and only if this is true of its left translation and right translation
If is a neighborhood basis of the identity element in a topological group then for all
is a neighborhood basis of in
In particular, any group topology on a topological group is completely determined by any neighborhood basis at the identity element.
If is any subset of and is an open subset of then is an open subset of

Symmetric neighborhoods

The inversion operation on a topological group is a homeomorphism from to itself.
A subset is said to be symmetric if where
The closure of every symmetric set in a commutative topological group is symmetric.
If is any subset of a commutative topological group, then the following sets are also symmetric:,, and.
For any neighborhood in a commutative topological group of the identity element, there exists a symmetric neighborhood of the identity element such that, where note that is necessarily a symmetric neighborhood of the identity element.
Thus every topological group has a neighborhood basis at the identity element consisting of symmetric sets.
If is a locally compact commutative group, then for any neighborhood in of the identity element, there exists a symmetric relatively compact neighborhood of the identity element such that .

Uniform space

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 is not abelian, then these two need not coincide.
The uniform structures allow one to talk about notions such as completeness, uniform continuity and uniform convergence on topological groups.

Separation properties

If is an open subset of a commutative topological group and contains a compact set, then there exists a neighborhood of the identity element such that.
As a uniform space, every commutative topological group is completely regular.
Consequently, for a multiplicative topological group with identity element 1, the following are equivalent:

is a T₀-space ;
is a T₂-space ;
is a T₃ ;
is closed in ;
, where is a neighborhood basis of the identity element in ;
for any such that there exists a neighborhood in of the identity element such that

A subgroup of a commutative topological group is discrete if and only if it has an isolated point.
If is not Hausdorff, then one can obtain a Hausdorff group by passing to the quotient group, where is the closure of the identity.
This is equivalent to taking the Kolmogorov quotient of.