In mathematics, topological groups are groups and topological spaces at the same time, where the group operations are required to be continuous. This connects these two structures together, relating them to each other.
Topological groups were studied extensively in the period of 1925 to 1940. Haar and Weil (respectively in 1933 and 1940) showed that the integrals and Fourier series are special cases of a construction 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 (in this case product):
and the inversion map:
are continuous.i.e. Continuous means that for any open set , is open in the domain of . 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 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.
Categorical definition of topological group
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 (binary product, unary inverse, and nullary identity), hence are categorical definitions.
Homomorphisms
A homomorphism of topological groups is defined to be 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.
For example, let be any group. We can put at least two topologies on it: the discrete topology or the indiscrete topology, both of which make the group operations continuous. Denoting the group with the discrete topology as , and the group with the indiscrete topology as , the identity map defined by is a continuous group isomorphism, but it is not an isomorphism of topological groups (if is not the trivial group).
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 (i.e. the trivial topology) also makes every group into a topological group.
The group of real numbers with the usual topology forms a topological group under addition. Euclidean -space is also a topological group under addition, and more generally, every topological vector space forms an (abelian) 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 (in fact, homeomorphic to the Cantor set), but it differs from (real) 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 is the class of 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 This makes every topological group into a homogeneous space. Consequently, for any and the subset is open (resp. closed) 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 If is any subset of a topological group , then the sets , , and are symmetric. For abelian , the closure of every symmetric set is symmetric.
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 (where is symmetric as well).
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 (Kolmogorov); is a T2-space (Hausdorff); is a T3 (Tychonoff); 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 .
Metrisability
Let be a topological group. As with any topological space, we say that is metrisable if and only if there exists a metric on , which induces the same topology on . A metric on is called
left-invariant (resp. right-invariant) if and only if (resp. ) for all (equivalently, is left-invariant just in case the map is an isometry from to itself for each ). proper if and only if all open balls, for , are pre-compact. The Birkhoff–Kakutani theorem (named after mathematicians Garrett Birkhoff and Shizuo Kakutani) states that the following three conditions on a topological group are equivalent:
is (Hausdorff and) first countable (equivalently: the identity element is closed in , and there is a countable basis of neighborhoods for in ). is metrisable (as a topological space). There is a left-invariant metric on that induces the given topology on . There is a right-invariant metric on that induces the given topology on .
Furthermore, the following are equivalent for any topological group :
is a second countable locally compact (Hausdorff) space. is a Polish, locally compact (Hausdorff) space. is properly metrisable (as a topological space). There is a left-invariant, proper metric on that induces the given topology on .
Note: As with the rest of the article we of assume here a Hausdorff topology. The implications 4 3 2 1 hold in any topological space. In particular 3 2 holds, since in particular any properly metrisable space is a countable union of compact metrisable, and thus separable (cf. properties of compact metric spaces), subsets.