TheInfoList

OR:

In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, topological groups are logically the combination of
groups A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...
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 have been studied extensively in the period of 1925 to 1940. Haar and Weil (respectively in 1933 and 1940) showed that the
integrals In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along with d ...
and
Fourier series A Fourier series () is a summation of harmonically related sinusoidal functions, also known as components or harmonics. The result of the summation is a periodic function whose functional form is determined by the choices of cycle length (or '' ...
are special cases of 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 Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear functions defi ...
, 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 In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called poin ...
that is also a group such that the group operation (in this case product): :, and the inversion map: :, are
continuous Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous g ...
.''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 In topology and related areas of mathematics, a product space is the Cartesian product of a family of topological spaces equipped with a natural topology called the product topology. This topology differs from another, perhaps more natural-seem ...
. 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 Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, ca ...
, topological groups can be defined concisely as
group object In category theory, a branch of mathematics, group objects are certain generalizations of groups that are built on more complicated structures than sets. A typical example of a group object is a topological group, a group whose underlying set is ...
s 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 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 In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isom ...
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 (i.e. the trivial topology) also makes every group into a topological group. The
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
s, $\mathbb$ 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 (abelian) topological group. Some other examples of
abelian Abelian may refer to: Mathematics Group theory * Abelian group, a group in which the binary operation is commutative ** Category of abelian groups (Ab), has abelian groups as objects and group homomorphisms as morphisms * Metabelian group, a grou ...
topological groups are the circle group , or the
torus In geometry, a torus (plural tori, colloquially donut or doughnut) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanar with the circle. If the axis of revolution does n ...
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 In mathematics, the orthogonal group in dimension , denoted , is the group of distance-preserving transformations of a Euclidean space of dimension that preserve a fixed point, where the group operation is given by composing transformations. ...
, the group of all
linear map In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that ...
s from to itself that preserve the
length Length is a measure of distance. In the International System of Quantities, length is a quantity with dimension distance. In most systems of measurement a base unit for length is chosen, from which all other units are derived. In the Inte ...
of all vectors. The orthogonal group is
compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in Britis ...
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 group In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the addi ...
s, 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 $\mathbb$ of
rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ra ...
s, with the topology inherited from $\mathbb$. This is a countable space, and it does not have the discrete topology. An important example for
number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Ma ...
is the group of ''p''-adic integers, for a
prime number A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime ...
, 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 is the locally compact field of ''p''-adic numbers. The group is a
pro-finite group In mathematics, a profinite group is a topological group that is in a certain sense assembled from a system of finite groups. The idea of using a profinite group is to provide a "uniform", or "synoptic", view of an entire system of finite groups. ...
; it is isomorphic to a subgroup of the product $\prod_ \mathbb / p^n$ in such a way that its topology is induced by the product topology, where the finite groups $\mathbb / p^n$ are given the discrete topology. Another large class of pro-finite 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 In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
, is an abelian topological group under addition. Some other infinite-dimensional groups that have been studied, with varying degrees of success, are
loop group In mathematics, a loop group is a group of loops in a topological group ''G'' with multiplication defined pointwise. Definition In its most general form a loop group is a group of continuous mappings from a manifold to a topological group . ...
s, Kac–Moody groups, diffeomorphism groups,
homeomorphism group In mathematics, particularly topology, the homeomorphism group of a topological space is the group consisting of all homeomorphisms from the space to itself with function composition as the group operation. Homeomorphism groups are very important ...
s, 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 $a \in G,$ left or right multiplication by this element yields a homeomorphism $G \to G.$ Consequently, for any $a \in G$ and $S \subseteq G,$ the subset $S$ is
open Open or OPEN may refer to: Music * Open (band), Australian pop/rock band * The Open (band), English indie rock band * Open (Blues Image album), ''Open'' (Blues Image album), 1969 * Open (Gotthard album), ''Open'' (Gotthard album), 1999 * Open (C ...
(resp.
closed Closed may refer to: Mathematics * Closure (mathematics), a set, along with operations, for which applying those operations on members always results in a member of the set * Closed set, a set which contains all its limit points * Closed interval, ...
) in $G$ if and only if this is true of its left translation $a S := \$ and right translation $S a := \.$ If $\mathcal$ is a neighborhood basis of the identity element in a topological group $G$ then for all $x \in X,$ $x \mathcal := \$ is a neighborhood basis of $x$ in $G.$ In particular, any group topology on a topological group is completely determined by any neighborhood basis at the identity element. If $S$ is any subset of $G$ and $U$ is an open subset of $G,$ then $S U := \$ is an open subset of $G.$

## Symmetric neighborhoods

The inversion operation $g \mapsto g^$ on a topological group $G$ is a homeomorphism from $G$ to itself. A subset $S \subseteq G$ is said to be symmetric if $S^ = S,$ where $S^ := \left\.$ 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 (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:
1. is a T0-space ( Kolmogorov);
2. is a T2-space ( Hausdorff);
3. is a T3 ( Tychonoff);
4. is closed in ;
5. , where is a neighborhood basis of the identity element in ;
6. for any $x \in G$ such that $x \neq 1,$ there exists a neighborhood in of the identity element such that $x \not\in U.$
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 In topology and related areas of mathematics, a metrizable space is a topological space that is homeomorphic to a metric space. That is, a topological space (X, \mathcal) is said to be metrizable if there is a metric d : X \times X \to , \inft ...
if and only if there exists a metric on , which induces the same topology on $G$. A metric on is called * ''left-invariant'' (resp. ''right-invariant'') if and only if $d\left(ax_,ax_\right)=d\left(x_,x_\right)$(resp. $d\left(x_a,x_a\right)=d\left(x_,x_\right)$) for all $a,x_,x_\in G$ (equivalently, $d$ is left-invariant just in case the map $x \mapsto ax$ is an
isometry In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective. The word isometry is derived from the Ancient Greek: ἴσος ''isos'' ...
from $\left(G,d\right)$ to itself for each $a \in G$). * ''proper'' if and only if all open balls, $B\left(r\right)=\$ for $r>0$, 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 first countable (equivalently: the identity element 1 is closed in , and there is a countable basis of neighborhoods for 1 in ). # is
metrisable In topology and related areas of mathematics, a metrizable space is a topological space that is homeomorphic to a metric space. That is, a topological space (X, \mathcal) is said to be metrizable if there is a metric d : X \times X \to , \inft ...
(as a topological space). # There is a left-invariant metric on that induces the given topology on . Furthermore, the following are equivalent for any topological group : # is a
second countable In topology, a second-countable space, also called a completely separable space, is a topological space whose topology has a countable base. More explicitly, a topological space T is second-countable if there exists some countable collection \ma ...
locally compact (Hausdorff) space. # is a Polish, locally compact (Hausdorff) space. # is properly
metrisable In topology and related areas of mathematics, a metrizable space is a topological space that is homeomorphic to a metric space. That is, a topological space (X, \mathcal) is said to be metrizable if there is a metric d : X \times X \to , \inft ...
(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 $\Rightarrow$ 3 $\Rightarrow$ 2 $\Rightarrow$ 1 hold in any topological space. In particular 3 $\Rightarrow$ 2 holds, since in particular any properly metrisable space is countable union of compact metrisable and thus separable (''cf.'' properties of compact metric spaces) subsets. The non-trivial implication 1 $\Rightarrow$ 4 was first proved by Raimond Struble in 1974. An alternative approach was made by Uffe Haagerup and Agata Przybyszewska in 2006, the idea of the which is as follows: One relies on the construction of a left-invariant metric, $d_$, as in the case of first countable spaces. By local compactness, closed balls of sufficiently small radii are compact, and by normalising we can assume this holds for radius 1. Closing the open ball, , of radius 1 under multiplication yields a clopen subgroup, , of , on which the metric $d_$ is proper. Since is open and is
second countable In topology, a second-countable space, also called a completely separable space, is a topological space whose topology has a countable base. More explicitly, a topological space T is second-countable if there exists some countable collection \ma ...
, the subgroup has at most countably many cosets. One now uses this sequence of cosets and the metric on to construct a proper metric on .

## Subgroups

Every subgroup of a topological group is itself a topological group when given the subspace topology. Every open subgroup is also closed in , since the complement of is the open set given by the union of open sets for . If is a subgroup of then the closure of is also a subgroup. Likewise, if is a normal subgroup of , the closure of is normal in .

## Quotients and normal subgroups

If is a subgroup of , the set of left cosets with the quotient topology is called a
homogeneous space In mathematics, particularly in the theories of Lie groups, algebraic groups and topological groups, a homogeneous space for a group ''G'' is a non-empty manifold or topological space ''X'' on which ''G'' acts transitively. The elements ...
for . The quotient map $q : G \to G / H$ is always
open Open or OPEN may refer to: Music * Open (band), Australian pop/rock band * The Open (band), English indie rock band * Open (Blues Image album), ''Open'' (Blues Image album), 1969 * Open (Gotthard album), ''Open'' (Gotthard album), 1999 * Open (C ...
. For example, for a positive integer , the
sphere A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the c ...
is a homogeneous space for the rotation group in , with . A homogeneous space is Hausdorff if and only if is closed in . Partly for this reason, it is natural to concentrate on closed subgroups when studying topological groups. If is a normal subgroup of , then the quotient group becomes a topological group when given the quotient topology. It is Hausdorff if and only if is closed in . For example, the quotient group is isomorphic to the circle group . In any topological group, the identity component (i.e., the connected component containing the identity element) is a closed normal subgroup. If is the identity component and ''a'' is any point of , then the left coset is the component of containing ''a''. So the collection of all left cosets (or right cosets) of in is equal to the collection of all components of . It follows that the quotient group is totally disconnected.

## Closure and compactness

In any commutative topological group, the product (assuming the group is multiplicative) of a compact set and a closed set is a closed set. Furthermore, for any subsets and of , . If is a subgroup of a commutative topological group and if is a neighborhood in of the identity element such that is closed, then is closed. Every discrete subgroup of a Hausdorff commutative topological group is closed.

## Isomorphism theorems

The
isomorphism theorem In mathematics, specifically abstract algebra, the isomorphism theorems (also known as Noether's isomorphism theorems) are theorems that describe the relationship between quotients, homomorphisms, and subobjects. Versions of the theorems exis ...
s from ordinary group theory are not always true in the topological setting. This is because a bijective homomorphism need not be an isomorphism of topological groups. For example, a native version of the first isomorphism theorem is false for topological groups: if $f:G\to H$ is a morphism of topological groups (that is, a continuous homomorphism), it is not necessarily true that the induced homomorphism $\tilde :G/\ker f\to \mathrm\left(f\right)$ is an isomorphism of topological groups; it will be a bijective, continuous homomorphism, but it will not necessarily be a homeomorphism. In other words, it will not necessarily admit an inverse in the category of topological groups. There is a version of the first isomorphism theorem for topological groups, which may be stated as follows: if $f : G \to H$ is a continuous homomorphism, then the induced homomorphism from to is an isomorphism if and only if the map is open onto its image. The third isomorphism theorem, however, is true more or less verbatim for topological groups, as one may easily check.

# Hilbert's fifth problem

There are several strong results on the relation between topological groups and Lie groups. First, every continuous homomorphism of Lie groups $G \to H$ is smooth. It follows that a topological group has a unique structure of a Lie group if one exists. Also, Cartan's theorem says that every closed subgroup of a Lie group is a Lie subgroup, in particular a smooth submanifold.
Hilbert's fifth problem Hilbert's fifth problem is the fifth mathematical problem from the problem list publicized in 1900 by mathematician David Hilbert, and concerns the characterization of Lie groups. The theory of Lie groups describes continuous symmetry in mathema ...
asked whether a topological group that is a topological manifold must be a Lie group. In other words, does have the structure of a smooth manifold, making the group operations smooth? As shown by Andrew Gleason, Deane Montgomery, and Leo Zippin, the answer to this problem is yes. In fact, has a
real analytic In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions. Functions of each type are infinitely differentiable, but complex a ...
structure. Using the smooth structure, one can define the Lie algebra of , an object of
linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrice ...
that determines a connected group up to covering spaces. As a result, the solution to Hilbert's fifth problem reduces the classification of topological groups that are topological manifolds to an algebraic problem, albeit a complicated problem in general. The theorem also has consequences for broader classes of topological groups. First, every compact group (understood to be Hausdorff) is an inverse limit of compact Lie groups. (One important case is an inverse limit of finite groups, called a profinite group. For example, the group of ''p''-adic integers and the absolute Galois group of a field are profinite groups.) Furthermore, every connected locally compact group is an inverse limit of connected Lie groups. At the other extreme, a totally disconnected locally compact group always contains a compact open subgroup, which is necessarily a profinite group. (For example, the locally compact group contains the compact open subgroup , which is the inverse limit of the finite groups as ' goes to infinity.)

# Representations of compact or locally compact groups

An action of a topological group on a topological space ''X'' is a group action of on ''X'' such that the corresponding function is continuous. Likewise, a representation of a topological group on a real or complex topological vector space ''V'' is a continuous action of on ''V'' such that for each , the map from ''V'' to itself is linear. Group actions and representation theory are particularly well understood for compact groups, generalizing what happens for finite groups. For example, every finite-dimensional (real or complex) representation of a compact group is a direct sum of irreducible representations. An infinite-dimensional unitary representation of a compact group can be decomposed as a Hilbert-space direct sum of irreducible representations, which are all finite-dimensional; this is part of the Peter–Weyl theorem. For example, the theory of
Fourier series A Fourier series () is a summation of harmonically related sinusoidal functions, also known as components or harmonics. The result of the summation is a periodic function whose functional form is determined by the choices of cycle length (or '' ...
describes the decomposition of the unitary representation of the circle group on the complex Hilbert space . The irreducible representations of are all 1-dimensional, of the form for integers (where is viewed as a subgroup of the multiplicative group *). Each of these representations occurs with multiplicity 1 in . The irreducible representations of all compact connected Lie groups have been classified. In particular, the
character Character or Characters may refer to: Arts, entertainment, and media Literature * ''Character'' (novel), a 1936 Dutch novel by Ferdinand Bordewijk * ''Characters'' (Theophrastus), a classical Greek set of character sketches attributed to The ...
of each irreducible representation is given by the Weyl character formula. More generally, locally compact groups have a rich theory of harmonic analysis, because they admit a natural notion of measure and
integral In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along with ...
, given by the Haar measure. Every unitary representation of a locally compact group can be described as a direct integral of irreducible unitary representations. (The decomposition is essentially unique if is of Type I, which includes the most important examples such as abelian groups and semisimple Lie groups.) A basic example is the
Fourier transform A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed ...
, which decomposes the action of the additive group $\mathbb$ on the Hilbert space as a direct integral of the irreducible unitary representations of $\mathbb$. The irreducible unitary representations of $\mathbb$ are all 1-dimensional, of the form for . The irreducible unitary representations of a locally compact group may be infinite-dimensional. A major goal of representation theory, related to the
Langlands classification In mathematics, the Langlands classification is a description of the irreducible representations of a reductive Lie group ''G'', suggested by Robert Langlands (1973). There are two slightly different versions of the Langlands classification. One ...
of admissible representations, is to find the unitary dual (the space of all irreducible unitary representations) for the semisimple Lie groups. The unitary dual is known in many cases such as , but not all. For a locally compact abelian group , every irreducible unitary representation has dimension 1. In this case, the unitary dual $\hat$ is a group, in fact another locally compact abelian group. Pontryagin duality states that for a locally compact abelian group , the dual of $\hat$ is the original group . For example, the dual group of the integers is the circle group , while the group $\mathbb$ of real numbers is isomorphic to its own dual. Every locally compact group has a good supply of irreducible unitary representations; for example, enough representations to distinguish the points of (the Gelfand–Raikov theorem). By contrast, representation theory for topological groups that are not locally compact has so far been developed only in special situations, and it may not be reasonable to expect a general theory. For example, there are many abelian Banach–Lie groups for which every representation on Hilbert space is trivial.

# Homotopy theory of topological groups

Topological groups are special among all topological spaces, even in terms of their homotopy type. One basic point is that a topological group determines a path-connected topological space, the classifying space (which classifies principal -bundles over topological spaces, under mild hypotheses). The group is isomorphic in the homotopy category to the
loop space In topology, a branch of mathematics, the loop space Ω''X'' of a pointed topological space ''X'' is the space of (based) loops in ''X'', i.e. continuous pointed maps from the pointed circle ''S''1 to ''X'', equipped with the compact-open topol ...
of ; that implies various restrictions on the homotopy type of . Some of these restrictions hold in the broader context of
H-space In mathematics, an H-space is a homotopy-theoretic version of a generalization of the notion of topological group, in which the axioms on associativity and inverses are removed. Definition An H-space consists of a topological space , together w ...
s. For example, the fundamental group of a topological group is abelian. (More generally, the Whitehead product on the homotopy groups of is zero.) Also, for any field ''k'', the
cohomology In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be view ...
ring has the structure of a Hopf algebra. In view of structure theorems on Hopf algebras by
Heinz Hopf Heinz Hopf (19 November 1894 – 3 June 1971) was a German mathematician who worked on the fields of topology and geometry. Early life and education Hopf was born in Gräbschen, Germany (now , part of Wrocław, Poland), the son of Eliza ...
and
Armand Borel Armand Borel (21 May 1923 – 11 August 2003) was a Swiss mathematician, born in La Chaux-de-Fonds, and was a permanent professor at the Institute for Advanced Study in Princeton, New Jersey, United States from 1957 to 1993. He worked in ...
, this puts strong restrictions on the possible cohomology rings of topological groups. In particular, if is a path-connected topological group whose rational cohomology ring is finite-dimensional in each degree, then this ring must be a free graded-commutative algebra over $\mathbb$, that is, the
tensor product In mathematics, the tensor product V \otimes W of two vector spaces and (over the same field) is a vector space to which is associated a bilinear map V\times W \to V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an element of V \otime ...
of a
polynomial ring In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring (which is also a commutative algebra) formed from the set of polynomials in one or more indeterminates (traditionally also called variables ...
on generators of even degree with an exterior algebra on generators of odd degree. In particular, for a connected Lie group , the rational cohomology ring of is an exterior algebra on generators of odd degree. Moreover, a connected Lie group has a maximal compact subgroup ''K'', which is unique up to conjugation, and the inclusion of ''K'' into is a
homotopy equivalence In topology, a branch of mathematics, two continuous functions from one topological space to another are called homotopic (from grc, ὁμός "same, similar" and "place") if one can be "continuously deformed" into the other, such a defo ...
. So describing the homotopy types of Lie groups reduces to the case of compact Lie groups. For example, the maximal compact subgroup of is the circle group , and the homogeneous space can be identified with the
hyperbolic plane In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai– Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with: :For any given line ''R'' and point ' ...
. Since the hyperbolic plane is
contractible In mathematics, a topological space ''X'' is contractible if the identity map on ''X'' is null-homotopic, i.e. if it is homotopic to some constant map. Intuitively, a contractible space is one that can be continuously shrunk to a point within th ...
, the inclusion of the circle group into is a homotopy equivalence. Finally, compact connected Lie groups have been classified by Wilhelm Killing,
Élie Cartan Élie Joseph Cartan (; 9 April 1869 – 6 May 1951) was an influential French mathematician who did fundamental work in the theory of Lie groups, differential systems (coordinate-free geometric formulation of PDEs), and differential geometr ...
, and Hermann Weyl. As a result, there is an essentially complete description of the possible homotopy types of Lie groups. For example, a compact connected Lie group of dimension at most 3 is either a torus, the group SU(2) ( diffeomorphic to the 3-sphere ), or its quotient group (diffeomorphic to ).

# Complete topological group

Information about convergence of nets and filters, such as definitions and properties, can be found in the article about filters in topology.

## Canonical uniformity on a commutative topological group

This article will henceforth assume that any topological group that we consider is an additive commutative topological group with identity element $0.$ The diagonal of $X$ is the set $\Delta_X := \$ and for any $N \subseteq X$ containing $0,$ the canonical entourage or canonical vicinities around $N$ is the set For a topological group $\left(X, \tau\right),$ the canonical uniformity on $X$ is the uniform structure induced by the set of all canonical entourages $\Delta\left(N\right)$ as $N$ ranges over all neighborhoods of $0$ in $X.$ That is, it is the upward closure of the following prefilter on $X \times X,$ $\left\$ where this prefilter forms what is known as a base of entourages of the canonical uniformity. For a commutative additive group $X,$ a fundamental system of entourages $\mathcal$ is called a translation-invariant uniformity if for every $B \in \mathcal,$ $\left(x, y\right) \in B$ if and only if $\left(x + z, y + z\right) \in B$ for all $x, y, z \in X.$ A uniformity $\mathcal$ is called translation-invariant if it has a base of entourages that is translation-invariant.
• The canonical uniformity on any commutative topological group is translation-invariant.
• The same canonical uniformity would result by using a neighborhood basis of the origin rather the filter of all neighborhoods of the origin.
• Every entourage $\Delta_X\left(N\right)$ contains the diagonal $\Delta_X := \Delta_X\left(\\right) = \$ because $0 \in N.$
• If $N$ is symmetric (that is, $-N = N$) then $\Delta_X\left(N\right)$ is symmetric (meaning that $\Delta_X\left(N\right)^ = \Delta_X\left(N\right)$) and
• The topology induced on $X$ by the canonical uniformity is the same as the topology that $X$ started with (that is, it is $\tau$).

## Cauchy prefilters and nets

The general theory of uniform spaces has its own definition of a "Cauchy prefilter" and "Cauchy net." For the canonical uniformity on $X,$ these reduces down to the definition described below. Suppose $x_ = \left\left(x_i\right\right)_$ is a net in $X$ and $y_ = \left\left(y_j\right\right)_$ is a net in $Y.$ Make $I \times J$ into a directed set by declaring $\left(i, j\right) \leq \left\left(i_2, j_2\right\right)$ if and only if $i \leq i_2 \text j \leq j_2.$ Then $x_ \times y_: = \left\left(x_i, y_j\right\right)_$ denotes the product net. If $X = Y$ then the image of this net under the addition map $X \times X \to X$ denotes the sum of these two nets: $x_ + y_: = \left(x_i + y_j\right)_$ and similarly their difference is defined to be the image of the product net under the subtraction map: $x_ - y_: = \left(x_i - y_j\right)_.$ A net $x_ = \left\left(x_i\right\right)_$ in an additive topological group $X$ is called a Cauchy net if $\left(x_i - x_j\right)_ \to 0 \text X$ or equivalently, if for every neighborhood $N$ of $0$ in $X,$ there exists some $i_0 \in I$ such that $x_i - x_j \in N$ for all indices $i, j \geq i_0.$ A Cauchy sequence is a Cauchy net that is a sequence. If $B$ is a subset of an additive group $X$ and $N$ is a set containing $0,$ then$B$ is said to be an $N$-small set or small of order $N$ if $B - B \subseteq N.$ A prefilter $\mathcal$ on an additive topological group $X$ called a Cauchy prefilter if it satisfies any of the following equivalent conditions:
1. $\mathcal - \mathcal \to 0$ in $X,$ where $\mathcal - \mathcal := \$ is a prefilter.
2. $\ \to 0$ in $X,$ where $\$ is a prefilter equivalent to $\mathcal - \mathcal.$
3. For every neighborhood $N$ of $0$ in $X,$ $\mathcal$ contains some $N$-small set (that is, there exists some $B \in \mathcal$ such that $B - B \subseteq N$).
and if $X$ is commutative then also:
1. For every neighborhood $N$ of $0$ in $X,$ there exists some $B \in \mathcal$ and some $x \in X$ such that $B \subseteq x + N.$
* It suffices to check any of the above condition for any given neighborhood basis of $0$ in $X.$ Suppose $\mathcal$ is a prefilter on a commutative topological group $X$ and $x \in X.$ Then$\mathcal \to x$ in $X$ if and only if $x \in \operatorname \mathcal$ and $\mathcal$ is Cauchy.

## Complete commutative topological group

Recall that for any $S \subseteq X,$ a prefilter $\mathcal$ ''on $S$'' is necessarily a subset of $\wp\left(S\right)$; that is, $\mathcal \subseteq \wp\left(S\right).$ A subset $S$ of a topological group $X$ is called a complete subset if it satisfies any of the following equivalent conditions:
1. Every Cauchy prefilter $\mathcal \subseteq \wp\left(S\right)$ on $S$ converges to at least one point of $S.$ * If $X$ is Hausdorff then every prefilter on $S$ will converge to at most one point of $X.$ But if $X$ is not Hausdorff then a prefilter may converge to multiple points in $X.$ The same is true for nets.
2. Every Cauchy net in $S$ converges to at least one point of $S$;
3. Every Cauchy filter $\mathcal$ on $S$ converges to at least one point of $S.$
4. $S$ is a
complete Complete may refer to: Logic * Completeness (logic) * Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable Mathematics * The completeness of the real numbers, which implies t ...
uniform space (under the point-set topology definition of "
complete uniform space In the mathematical field of topology, a uniform space is a set with a uniform structure. Uniform spaces are topological spaces with additional structure that is used to define uniform properties such as completeness, uniform continuity and unif ...
") when $S$ is endowed with the uniformity induced on it by the canonical uniformity of $X$;
A subset $S$ is called a sequentially complete subset if every Cauchy sequence in $S$ (or equivalently, every elementary Cauchy filter/prefilter on $S$) converges to at least one point of $S.$ * Importantly, convergence outside of $S$ is allowed: If $X$ is not Hausdorff and if every Cauchy prefilter on $S$ converges to some point of $S,$ then $S$ will be complete even if some or all Cauchy prefilters on $S$ ''also'' converge to points(s) in the complement $X \setminus S.$ In short, there is no requirement that these Cauchy prefilters on $S$ converge ''only'' to points in $S.$ The same can be said of the convergence of Cauchy nets in $S.$ ** As a consequence, if a commutative topological group $X$ is ''not'' Hausdorff, then every subset of the closure of $\,$ say $S \subseteq \operatorname \,$ is complete (since it is clearly compact and every compact set is necessarily complete). So in particular, if $S \neq \varnothing$ (for example, if $S$ a is singleton set such as $S = \$) then $S$ would be complete even though ''every'' Cauchy net in $S$ (and every Cauchy prefilter on $S$), converges to ''every'' point in $\operatorname \$ (include those points in $\operatorname \$ that are not in $S$). ** This example also shows that complete subsets (indeed, even compact subsets) of a non-Hausdorff space may fail to be closed (for example, if $\varnothing \neq S \subseteq \operatorname \$ then $S$ is closed if and only if $S = \operatorname \$). A commutative topological group $X$ is called a complete group if any of the following equivalent conditions hold:
1. $X$ is complete as a subset of itself.
2. Every Cauchy net in $X$ converges to at least one point of $X.$
3. There exists a neighborhood of $0$ in $X$ that is also a complete subset of $X.$ * This implies that every locally compact commutative topological group is complete.
4. When endowed with its canonical uniformity, $X$ becomes is a
complete uniform space In the mathematical field of topology, a uniform space is a set with a uniform structure. Uniform spaces are topological spaces with additional structure that is used to define uniform properties such as completeness, uniform continuity and unif ...
. * In the general theory of uniform spaces, a uniform space is called a
complete uniform space In the mathematical field of topology, a uniform space is a set with a uniform structure. Uniform spaces are topological spaces with additional structure that is used to define uniform properties such as completeness, uniform continuity and unif ...
if each Cauchy filter in $X$ converges in $\left(X, \tau\right)$ to some point of $X.$
A topological group is called sequentially complete if it is a sequentially complete subset of itself. Neighborhood basis: Suppose $C$ is a completion of a commutative topological group $X$ with $X \subseteq C$ and that $\mathcal$ is a neighborhood base of the origin in $X.$ Then the family of sets $\left\$ is a neighborhood basis at the origin in $C.$ Let $X$ and $Y$ be topological groups, $D \subseteq X,$ and $f : D \to Y$ be a map. Then $f : D \to Y$ is uniformly continuous if for every neighborhood $U$ of the origin in $X,$ there exists a neighborhood $V$ of the origin in $Y$ such that for all $x, y \in D,$ if $y - x \in U$ then $f\left(y\right) - f\left(x\right) \in V.$

# Generalizations

Various generalizations of topological groups can be obtained by weakening the continuity conditions: * A semitopological group is a group with a topology such that for each the two functions defined by and are continuous. * A
quasitopological group In mathematics, topological groups are logically 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 st ...
is a semitopological group in which the function mapping elements to their inverses is also continuous. * A paratopological group is a group with a topology such that the group operation is continuous.