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In 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 di ...
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 ''p ...
are special cases of a very wide class of topological groups. Topological groups, along with continuous group actions, are used to study continuous
symmetries Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definiti ...
, 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 defined on ...
, every
topological vector space In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis. A topological vector space is a vector space that is al ...
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 points ...
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 ...
.''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-seemi ...
. 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, categ ...
, 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 isomorp ...
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 In topology, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points form a , meaning they are '' isolated'' from each other in a certain sense. The discrete topology is the finest t ...
; 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 numbers, $\mathbb$ 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 In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis. A topological vector space is a vector space that is al ...
forms an (abelian) topological group. Some other examples of abelian 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 no ...
for any natural number . The
classical group In mathematics, the classical groups are defined as the special linear groups over the reals , the complex numbers and the quaternions together with special automorphism groups of symmetric or skew-symmetric bilinear forms and Hermitian or ...
s are important examples of non-abelian topological groups. For instance, the general linear group of all invertible -by-
matrices Matrix most commonly refers to: * ''The Matrix'' (franchise), an American media franchise ** ''The Matrix'', a 1999 science-fiction action film ** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...
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. T ...
, 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 pre ...
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 Inter ...
of all vectors. The orthogonal group is compact as a topological space. Much of
Euclidean geometry Euclidean geometry is a mathematical system attributed to ancient Greek mathematician Euclid, which he described in his textbook on geometry: the '' Elements''. Euclid's approach consists in assuming a small set of intuitively appealing axiom ...
can be viewed as studying the structure of the orthogonal group, or the closely related group of
isometries 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'' mea ...
of . The groups mentioned so far are all Lie groups, meaning that they are
smooth manifold In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One ma ...
s in such a way that the group operations are
smooth Smooth may refer to: Mathematics * Smooth function, a function that is infinitely differentiable; used in calculus and topology * Smooth manifold, a differentiable manifold for which all the transition maps are smooth functions * Smooth algebraic ...
, 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 algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi identi ...
s and then solved. An example of a topological group that is not a Lie group is the additive group $\mathbb$ of rational numbers, 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 is the group of ''p''-adic integers, for a prime number , meaning the
inverse limit In mathematics, the inverse limit (also called the projective limit) is a construction that allows one to "glue together" several related objects, the precise gluing process being specified by morphisms between the objects. Thus, inverse limits ca ...
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 group In mathematics, a locally compact group is a topological group ''G'' for which the underlying topology is locally compact and Hausdorff. Locally compact groups are important because many examples of groups that arise throughout mathematics are loc ...
s such as , where is the locally compact field of ''p''-adic numbers. The group is a pro-finite group; 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 In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis. A topological vector space is a vector space that is al ...
, 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 naturall ...
, 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 In mathematics, especially functional analysis, a Banach algebra, named after Stefan Banach, is an associative algebra A over the real or complex numbers (or over a non-Archimedean complete normed field) that at the same time is also a Banach ...
with multiplicative identity, the set of invertible elements forms a topological group under multiplication. For example, the group of invertible
bounded operator In functional analysis and operator theory, a bounded linear operator is a linear transformation L : X \to Y between topological vector spaces (TVSs) X and Y that maps bounded subsets of X to bounded subsets of Y. If X and Y are normed vector ...
s 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), 1969 * ''Open'' (Gotthard album), 1999 * ''Open'' (Cowboy Junkies album), 2001 * ''Open'' ( ...
(resp. closed) 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 Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definiti ...
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 In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. More precisely, it is a topological space in which ev ...
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 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 un ...
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 In mathematics, a real function f of real numbers is said to be uniformly continuous if there is a positive real number \delta such that function values over any function domain interval of the size \delta are as close to each other as we want. In ...
and
uniform convergence In the mathematical field of analysis, uniform convergence is a mode of convergence of functions stronger than pointwise convergence. A sequence of functions (f_n) converges uniformly to a limiting function f on a set E if, given any arbitrarily ...
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 Andrey Nikolaevich Kolmogorov ( rus, Андре́й Никола́евич Колмого́ров, p=ɐnˈdrʲej nʲɪkɐˈlajɪvʲɪtɕ kəlmɐˈɡorəf, a=Ru-Andrey Nikolaevich Kolmogorov.ogg, 25 April 1903 – 20 October 1987) was a Sovi ...
);
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 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 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 was a Japanese-American mathematician, best known for his eponymous fixed-point theorem. Biography Kakutani attended Tohoku University in Sendai, where his advisor was Tatsujirō Shimizu. At one point he spent two years at the Institute for ...
) 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 (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
locally compact In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. More precisely, it is a topological space in which ev ...
(Hausdorff) space. # is a Polish,
locally compact In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. More precisely, it is a topological space in which ev ...
(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 $\Rightarrow$ 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 In topology, a clopen set (a portmanteau of closed-open set) in a topological space is a set which is both open and closed. That this is possible may seem counter-intuitive, as the common meanings of and are antonyms, but their mathematical ...
subgroup, , of , on which the metric $d_$ is proper. Since is open and is second countable, 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 In group theory, a branch of mathematics, given a group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely, ''H'' is a subgro ...
of a topological group is itself a topological group when given the
subspace topology In topology and related areas of mathematics, a subspace of a topological space ''X'' is a subset ''S'' of ''X'' which is equipped with a topology induced from that of ''X'' called the subspace topology (or the relative topology, or the induced to ...
. 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 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), 1969 * ''Open'' (Gotthard album), 1999 * ''Open'' (Cowboy Junkies album), 2001 * ''Open'' ( ...
. 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 ...
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 In mathematics, specifically group theory, the identity component of a group ''G'' refers to several closely related notions of the largest connected subgroup of ''G'' containing the identity element. In point set topology, the identity compon ...
(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 theorems 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 In mathematics, a submanifold of a manifold ''M'' is a subset ''S'' which itself has the structure of a manifold, and for which the inclusion map satisfies certain properties. There are different types of submanifolds depending on exactly which p ...
. Hilbert's fifth problem 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 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 matrices. ...
that determines a
connected Connected may refer to: Film and television * ''Connected'' (2008 film), a Hong Kong remake of the American movie ''Cellular'' * '' Connected: An Autoblogography About Love, Death & Technology'', a 2011 documentary film * ''Connected'' (2015 TV ...
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 Representation may refer to: Law and politics *Representation (politics), political activities undertaken by elected representatives, as well as other theories ** Representative democracy, type of democracy in which elected officials represent a ...
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 In mathematics, a unitary representation of a group ''G'' is a linear representation π of ''G'' on a complex Hilbert space ''V'' such that π(''g'') is a unitary operator for every ''g'' ∈ ''G''. The general theory is well-developed in case ''G ...
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 ''p ...
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 of each irreducible representation is given by the Weyl character formula. More generally, locally compact groups have a rich theory of
harmonic analysis Harmonic analysis is a branch of mathematics concerned with the representation of functions or signals as the superposition of basic waves, and the study of and generalization of the notions of Fourier series and Fourier transforms (i.e. an ex ...
, 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 dif ...
, given by the
Haar measure In mathematical analysis, the Haar measure assigns an "invariant volume" to subsets of locally compact topological groups, consequently defining an integral for functions on those groups. This measure was introduced by Alfréd Haar in 1933, thoug ...
. 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 group In mathematics, a Lie algebra is semisimple if it is a direct sum of simple Lie algebras. (A simple Lie algebra is a non-abelian Lie algebra without any non-zero proper ideals). Throughout the article, unless otherwise stated, a Lie algebra is ...
s.) A basic example is the Fourier transform, 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 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 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 deforma ...
. One basic point is that a topological group determines a path-connected topological space, the
classifying space In mathematics, specifically in homotopy theory, a classifying space ''BG'' of a topological group ''G'' is the quotient of a weakly contractible space ''EG'' (i.e. a topological space all of whose homotopy groups are trivial) by a proper free ac ...
(which classifies principal -bundles over topological spaces, under mild hypotheses). The group is isomorphic in the
homotopy category In mathematics, the homotopy category is a category built from the category of topological spaces which in a sense identifies two spaces that have the same shape. The phrase is in fact used for two different (but related) categories, as discussed be ...
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 topolo ...
of ; that implies various restrictions on the homotopy type of . Some of these restrictions hold in the broader context of H-spaces. For example, the
fundamental group In the mathematical field of algebraic topology, the fundamental group of a topological space is the group of the equivalence classes under homotopy of the loops contained in the space. It records information about the basic shape, or holes, ...
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 ring has the structure of a
Hopf algebra Hopf is a German surname. Notable people with the surname include: * Eberhard Hopf (1902–1983), Austrian mathematician * Hans Hopf (1916–1993), German tenor * Heinz Hopf (1894–1971), German mathematician * Heinz Hopf (actor) (1934–2001), Sw ...
. 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 Elizabeth ( ...
and Armand Borel, 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 \otimes ...
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 variabl ...
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 In mathematics, a maximal compact subgroup ''K'' of a topological group ''G'' is a subgroup ''K'' that is a compact space, in the subspace topology, and maximal amongst such subgroups. Maximal compact subgroups play an important role in the classi ...
''K'', which is unique up to conjugation, and the inclusion of ''K'' into is a homotopy equivalence. 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. 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 Wilhelm Karl Joseph Killing (10 May 1847 – 11 February 1923) was a German mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics is an area of knowledge that includes the topics of nu ...
, Élie Cartan, and
Hermann Weyl Hermann Klaus Hugo Weyl, (; 9 November 1885 – 8 December 1955) was a German mathematician, theoretical physicist and philosopher. Although much of his working life was spent in Zürich, Switzerland, and then Princeton, New Jersey, he is assoc ...
. 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) In mathematics, the special unitary group of degree , denoted , is the Lie group of unitary matrices with determinant 1. The more general unitary matrices may have complex determinants with absolute value 1, rather than real 1 in the speci ...
(
diffeomorphic In mathematics, a diffeomorphism is an isomorphism of smooth manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are differentiable. Definition Given tw ...
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 Filters in topology, a subfield of mathematics, can be used to study topological spaces and define all basic topological notions such a convergence, continuity, compactness, and more. Filters, which are special families of subsets of some given ...
.

## 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 Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definiti ...
(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 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 un ...
s 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 Net or net may refer to: Mathematics and physics * Net (mathematics), a filter-like topological generalization of a sequence * Net, a linear system of divisors of dimension 2 * Net (polyhedron), an arrangement of polygons that can be folded ...
$x_ = \left\left(x_i\right\right)_$ 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 uniform space (under the point-set topology definition of " complete uniform space") 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 general theory of
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 un ...
s, a uniform space is called a complete uniform space 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 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.