{"history":[],"article":{"id":"fbdc3f56-8272-408e-835b-b4267e314cdb","title":"Topological group","slug":"topological-group","content":":::summary\n:::\n\n# \nIn [mathematics](https://bliptext.com/articles/mathematics), **topological [groups](https://bliptext.com/articles/group-mathematics)** are the combination of [groups](https://bliptext.com/articles/group-mathematics) and [topological spaces](https://bliptext.com/articles/topological-space), i.e. they are [groups](https://bliptext.com/articles/group-mathematics) and [topological spaces](https://bliptext.com/articles/topological-space) at the same time, such that the [continuity](https://bliptext.com/articles/continuous-function) condition for the group operations connects these two structures together and consequently they are not independent from each other.\nTopological groups have been studied extensively in the period of 1925 to 1940.\n[Haar](https://bliptext.com/articles/alfr-d-haar) and [Weil](https://bliptext.com/articles/andr-weil) (respectively in 1933 and 1940) showed that the [integrals](https://bliptext.com/articles/integral) and [Fourier series](https://bliptext.com/articles/fourier-series) are special cases of a very wide class of topological groups.\nTopological groups, along with [continuous group actions](https://bliptext.com/articles/continuous-group-action), are used to study continuous [symmetries](https://bliptext.com/articles/symmetry), which have many applications, for example, [in physics](https://bliptext.com/articles/symmetry-physics).\nIn [functional analysis](https://bliptext.com/articles/functional-analysis), every [topological vector space](https://bliptext.com/articles/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](https://bliptext.com/articles/functional-analysis).\n# [The real numbers form a topological group under addition | upload.wikimedia.org/wikipedia/commons/d/d7/Real///number///line.svg]\n\n# Formal definition\nA **topological group**, G, is a [topological space](https://bliptext.com/articles/topological-space) that is also a group such that the group operation (in this case product):\n⋅ : G × G → G\n,\n(x, y) ↦ xy\nand the inversion map:\n−1 : G → G\n,\nx ↦ x−1\nare [continuous](https://bliptext.com/articles/continuous-function).\nHere\nG × G\nis viewed as a topological space with the [product topology](https://bliptext.com/articles/product-topology).\nSuch a topology is said to be **compatible with the group operations** and is called a **group topology**.\nThe product map is continuous if and only if for any\nx, y ∈ G\nand any neighborhood W of\nxy\nin G, there exist neighborhoods U of x and V of y in G such that\nU ⋅ V ⊆ W\n, where\nU ⋅ V := {u ⋅ v : u ∈ U, v ∈ V}\n. The inversion map is continuous if and only if for any\nx ∈ G\nand any neighborhood V of\nx−1\nin G, there exists a neighborhood U of x in G such that\nU−1 ⊆ V\n, where\nU−1 := { u−1 : u ∈ U }\n.\nTo show that a topology is compatible with the group operations, it suffices to check that the map\nG × G → G\n,\n(x, y) ↦ xy−1\nis continuous.\nExplicitly, this means that for any\nx, y ∈ G\nand any neighborhood W in G of\nxy−1\n, there exist neighborhoods U of x and V of y in G such that\nU ⋅ (V−1) ⊆ W\n.\nThis definition used notation for multiplicative groups;\nthe equivalent for additive groups would be that the following two operations are continuous:\n+ : G × G → G\n,\n(x, y) ↦ x + y\n− : G → G\n,\nx ↦ −x\n.\nAlthough not part of this definition, many authors require that the topology on G be [Hausdorff](https://bliptext.com/articles/hausdorff-space).\nOne reason for this is that any topological group can be canonically associated with a Hausdorff topological group by taking an appropriate canonical quotient;\nthis however, often still requires working with the original non-Hausdorff topological group.\nOther reasons, and some equivalent conditions, are discussed below.\nThis article will not assume that topological groups are necessarily Hausdorff.\nIn the language of [category theory](https://bliptext.com/articles/category-theory), topological groups can be defined concisely as [group objects](https://bliptext.com/articles/group-object) in the [category of topological spaces](https://bliptext.com/articles/category-of-topological-spaces), in the same way that ordinary groups are [group objects](https://bliptext.com/articles/group-object) in the [category of sets](https://bliptext.com/articles/category-of-sets).\nNote that the axioms are given in terms of the maps (binary product, unary inverse, and nullary identity), hence are categorical definitions.\n\n## Homomorphisms\nA **homomorphism** of topological groups means a continuous group **homomorphism**\nG → H\n. Topological groups, together with their homomorphisms, form a [category](https://bliptext.com/articles/category-theory).\nA group homomorphism between topological groups is continuous if and only if it is continuous at some point.\nAn **isomorphism** of topological groups is a group **isomorphism** that is also a [homeomorphism](https://bliptext.com/articles/homeomorphism) of the underlying topological spaces.\nThis is stronger than simply requiring a continuous group isomorphism—the inverse must also be continuous.\nThere are examples of topological groups that are isomorphic as ordinary groups but not as topological groups.\nIndeed, any non-discrete topological group is also a topological group when considered with the discrete topology.\nThe underlying groups are the same, but as topological groups there is not an isomorphism.\n\n# Examples\nEvery group can be trivially made into a topological group by considering it with the [discrete topology](https://bliptext.com/articles/discrete-topology); such groups are called [discrete groups](https://bliptext.com/articles/discrete-group).\nIn this sense, the theory of topological groups subsumes that of ordinary groups.\nThe [indiscrete topology](https://bliptext.com/articles/indiscrete-topology) (i.e. the trivial topology) also makes every group into a topological group.\nThe [real numbers](https://bliptext.com/articles/real-number), \\mathbb{R} with the usual topology form a topological group under addition.\n[Euclidean n-space](https://bliptext.com/articles/euclidean-space)\n\\mathbb{R}n\nis also a topological group under addition, and more generally, every [topological vector space](https://bliptext.com/articles/topological-vector-space) forms an (abelian) topological group.\nSome other examples of [abelian](https://bliptext.com/articles/abelian-group) topological groups are the [circle group](https://bliptext.com/articles/circle-group)\nS1\n, or the [torus](https://bliptext.com/articles/torus-group)\n(S1)n\nfor any natural number n.\nThe [classical groups](https://bliptext.com/articles/classical-group) are important examples of non-abelian topological groups.\nFor instance, the [general linear group](https://bliptext.com/articles/general-linear-group)\nGL(n,\\mathbb{R})\nof all invertible n-by-n [matrices](https://bliptext.com/articles/matrix-mathematics) with real entries can be viewed as a topological group with the topology defined by viewing\nGL(n,\\mathbb{R})\nas a [subspace](https://bliptext.com/articles/subspace-topology) of Euclidean space\n\\mathbb{R}n×n\n. Another classical group is the [orthogonal group](https://bliptext.com/articles/orthogonal-group)\nO(n)\n, the group of all [linear maps](https://bliptext.com/articles/linear-map) from\n\\mathbb{R}n\nto itself that preserve the [length](https://bliptext.com/articles/euclidean-distance) of all vectors.\nThe orthogonal group is [compact](https://bliptext.com/articles/compact-space) as a topological space.\nMuch of [Euclidean geometry](https://bliptext.com/articles/euclidean-geometry) can be viewed as studying the structure of the orthogonal group, or the closely related group\nO(n) ⋉ \\mathbb{R}n\nof [isometries](https://bliptext.com/articles/euclidean-group) of\n\\mathbb{R}n\n.\nThe groups mentioned so far are all [Lie groups](https://bliptext.com/articles/lie-group), meaning that they are [smooth manifolds](https://bliptext.com/articles/[smooth](https://bliptext.com/articles/smooth-function)-manifold) in such a way that the group operations are [smooth](https://bliptext.com/articles/smooth-function), not just continuous.\nLie groups are the best-understood topological groups; many questions about Lie groups can be converted to purely algebraic questions about [Lie algebras](https://bliptext.com/articles/lie-algebra) and then solved.\nAn example of a topological group that is not a Lie group is the additive group \\mathbb{Q} of [rational numbers](https://bliptext.com/articles/rational-number), with the topology inherited from \\mathbb{R}.\nThis is a [countable](https://bliptext.com/articles/countable) space, and it does not have the discrete topology.\nAn important example for [number theory](https://bliptext.com/articles/number-theory) is the group\n\\mathbb{Z}p\nof p-adic integers, for a [prime number](https://bliptext.com/articles/prime-number) p, meaning the [inverse limit](https://bliptext.com/articles/inverse-limit) of the finite groups\n\\mathbb{Z}/pn\nas n goes to infinity.\nThe group\n\\mathbb{Z}p\nis well behaved in that it is compact (in fact, homeomorphic to the [Cantor set](https://bliptext.com/articles/cantor-set)), but it differs from (real) Lie groups in that it is [totally disconnected](https://bliptext.com/articles/totally-disconnected-group).\nMore generally, there is a theory of p-adic Lie groups, including compact groups such as\nGL(n,\\mathbb{Z}p)\nas well as [locally compact groups](https://bliptext.com/articles/locally-compact-group) such as\nGL(n,\\mathbb{Q}p)\n, where\n\\mathbb{Q}p\nis the locally compact [field](https://bliptext.com/articles/field-mathematics) of p-adic numbers.\nThe group\n\\mathbb{Z}p\nis a [pro-finite group](https://bliptext.com/articles/pro-finite-group); it is isomorphic to a subgroup of the product in such a way that its topology is induced by the product topology, where the finite groups are given the discrete topology.\nAnother large class of pro-finite groups important in number theory are [absolute Galois groups](https://bliptext.com/articles/absolute-galois-group).\nSome topological groups can be viewed as [infinite dimensional Lie groups](https://bliptext.com/articles/infinite-dimensional-lie-group); this phrase is best understood informally, to include several different families of examples.\nFor example, a [topological vector space](https://bliptext.com/articles/topological-vector-space), such as a [Banach space](https://bliptext.com/articles/banach-space) or [Hilbert space](https://bliptext.com/articles/hilbert-space), is an abelian topological group under addition.\nSome other infinite-dimensional groups that have been studied, with varying degrees of success, are [loop groups](https://bliptext.com/articles/loop-group), [Kac–Moody groups](https://bliptext.com/articles/kac-moody-algebra), [Diffeomorphism groups](https://bliptext.com/articles/diffeomorphism), [homeomorphism groups](https://bliptext.com/articles/homeomorphism-group), and [gauge groups](https://bliptext.com/articles/gauge-group).\nIn every [Banach algebra](https://bliptext.com/articles/banach-algebra) with multiplicative identity, the set of invertible elements forms a topological group under multiplication.\nFor example, the group of invertible [bounded operators](https://bliptext.com/articles/bounded-operator) on a Hilbert space arises this way.\n\n# Properties\n\n## Translation invariance\nEvery 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 the subset S is [open](https://bliptext.com/articles/open-set) (resp. [closed](https://bliptext.com/articles/closed-set)) in G if and only if this is true of its left translation and right translation\nIf \\mathcal{N} is a [neighborhood basis](https://bliptext.com/articles/neighborhood-basis) of the identity element in a topological group G then for all x \\in X,\nis 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.\nIf S is any subset of G and U is an open subset of G, then is an open subset of G.\n\n## Symmetric neighborhoods\nThe inversion operation on a topological group G is a homeomorphism from G to itself.\nA subset is said to be [symmetric](https://bliptext.com/articles/symmetric-set) if S^{-1} = S, where\nThe closure of every symmetric set in a commutative topological group is symmetric.\nIf S is any subset of a commutative topological group G, then the following sets are also symmetric:\nS−1 ∩ S\n,\nS−1 ∪ S\n, and\nS−1 S\n.\nFor any neighborhood N in a commutative topological group G of the identity element, there exists a symmetric neighborhood M of the identity element such that\nM−1 M ⊆ N\n, where note that\nM−1 M\nis necessarily a symmetric neighborhood of the identity element.\nThus every topological group has a neighborhood basis at the identity element consisting of symmetric sets.\nIf G is a [locally compact](https://bliptext.com/articles/locally-compact) commutative group, then for any neighborhood N in G of the identity element, there exists a symmetric relatively compact neighborhood M of the identity element such that\ncl M ⊆ N\n(where\ncl M\nis symmetric as well).\n\n## Uniform space\nEvery topological group can be viewed as a [uniform space](https://bliptext.com/articles/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.\nIf G is not abelian, then these two need not coincide.\nThe uniform structures allow one to talk about notions such as [completeness](https://bliptext.com/articles/completeness-topology), [uniform continuity](https://bliptext.com/articles/uniformly-continuous) and [uniform convergence](https://bliptext.com/articles/uniform-convergence) on topological groups.\n\n## Separation properties\nIf U is an open subset of a commutative topological group G and U contains a compact set K, then there exists a neighborhood N of the identity element such that\nKN ⊆ U\n.\nAs a uniform space, every commutative topological group is [completely regular](https://bliptext.com/articles/completely-regular-space).\nConsequently, for a multiplicative topological group G with identity element 1, the following are equivalent:\n

G is a T0-space ([Kolmogorov](https://bliptext.com/articles/kolmogorov-space));
G is a T2-space ([Hausdorff](https://bliptext.com/articles/hausdorff-space));
G is a T31/2 ([Tychonoff](https://bliptext.com/articles/tychonoff-space));
\n{ 1 }\nis closed in G;
\n{ 1 } := ∩ N ∈ 𝒩 N\n, where\n𝒩\nis a neighborhood basis of the identity element in G;
for any x \\in G such that x \\neq 1, there exists a neighborhood U in G of the identity element such that

\nA subgroup of a commutative topological group is discrete if and only if it has an [isolated point](https://bliptext.com/articles/isolated-point).\nIf G is not Hausdorff, then one can obtain a Hausdorff group by passing to the quotient group\nG/K\n, where K is the [closure](https://bliptext.com/articles/closure-topology) of the identity.\nThis is equivalent to taking the [Kolmogorov quotient](https://bliptext.com/articles/kolmogorov-space) of G.\n\n## Metrisability\nLet G be a topological group.\nAs with any topological space, we say that G is [metrisable](https://bliptext.com/articles/metrizable-space) if and only if there exists a metric d on G, which induces the same topology on G. A metric d on G is called\nThe **Birkhoff–Kakutani theorem** (named after mathematicians [Garrett Birkhoff](https://bliptext.com/articles/garrett-birkhoff) and [Shizuo Kakutani](https://bliptext.com/articles/shizuo-kakutani)) states that the following three conditions on a topological group G are equivalent:\nFurthermore, the following are equivalent for any topological group G:\n**Note:** As with the rest of the article we of assume here a Hausdorff topology.\nThe implications 4 \\Rightarrow 3 \\Rightarrow 2 \\Rightarrow 1 hold in any topological space.\nIn 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](https://bliptext.com/articles/compact-space)) subsets.\nThe non-trivial implication 1 \\Rightarrow 4 was first proved by Raimond Struble in 1974.\nAn alternative approach was made by [Uffe Haagerup](https://bliptext.com/articles/uffe-haagerup) and Agata Przybyszewska in 2006,\nthe idea of the which is as follows:\nOne relies on the construction of a left-invariant metric, d_{0}, as in the case of [first countable spaces](https://bliptext.com/articles/first-countable-space).\nBy local compactness, closed balls of sufficiently small radii are compact, and by normalising we can assume this holds for radius 1.\nClosing the open ball, U, of radius 1 under multiplication yields a [clopen](https://bliptext.com/articles/clopen) subgroup, H, of G, on which the metric d_{0} is proper.\nSince H is open and G is [second countable](https://bliptext.com/articles/second-countable-space), the subgroup has at most countably many cosets.\nOne now uses this sequence of cosets and the metric on H to construct a proper metric on G.\n\n## Subgroups\nEvery [subgroup](https://bliptext.com/articles/subgroup) of a topological group is itself a topological group when given the [subspace topology](https://bliptext.com/articles/subspace-topology).\nEvery open subgroup H is also closed in G, since the complement of H is the open set given by the union of open sets\ngH\nfor\ng ∈ G \\ H\n. If H is a subgroup of G then the closure of H is also a subgroup.\nLikewise, if H is a normal subgroup of G, the closure of H is normal in G.\n\n## Quotients and normal subgroups\nIf H is a subgroup of G, the set of left [cosets](https://bliptext.com/articles/coset)\nG/H\nwith the [quotient topology](https://bliptext.com/articles/quotient-topology) is called a [homogeneous space](https://bliptext.com/articles/homogeneous-space) for G. The quotient map is always [open](https://bliptext.com/articles/open-map).\nFor example, for a positive integer n, the [sphere](https://bliptext.com/articles/n-sphere)\nSn\nis a homogeneous space for the [rotation group](https://bliptext.com/articles/rotation-group)\nSO(n+1)\nin\n\\Rn+1\n, with\n. A homogeneous space\nG/H\nis Hausdorff if and only if H is closed in G. Partly for this reason, it is natural to concentrate on closed subgroups when studying topological groups.\nIf H is a [normal subgroup](https://bliptext.com/articles/normal-subgroup) of G, then the [quotient group](https://bliptext.com/articles/quotient-group)\nG/H\nbecomes a topological group when given the quotient topology.\nIt is Hausdorff if and only if H is closed in G. For example, the quotient group\n\\R / \\Z\nis isomorphic to the circle group\nS1\n.\nIn any topological group, the [identity component](https://bliptext.com/articles/identity-component) (i.e., the [connected component](https://bliptext.com/articles/connected-component-topology) containing the identity element) is a closed normal subgroup.\nIf C is the identity component and a is any point of G, then the left coset\naC\nis the component of G containing a.\nSo the collection of all left cosets (or right cosets) of C in G is equal to the collection of all components of G. It follows that the quotient group\nG/C\nis [totally disconnected](https://bliptext.com/articles/totally-disconnected).\n\n## Closure and compactness\nIn any commutative topological group, the product (assuming the group is multiplicative)\nKC\nof a compact set K and a closed set C is a closed set.\nFurthermore, for any subsets R and S of G,\n(cl R)(cl S) ⊆ cl (RS)\n.\nIf H is a subgroup of a commutative topological group G and if N is a neighborhood in G of the identity element such that H ∩ cl N is closed, then H is closed.\nEvery discrete subgroup of a Hausdorff commutative topological group is closed.\n\n## Isomorphism theorems\nThe [isomorphism theorems](https://bliptext.com/articles/isomorphism-theorem) from ordinary group theory are not always true in the topological setting.\nThis is because a bijective homomorphism need not be an isomorphism of topological groups.\nFor 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 is an isomorphism of topological groups; it will be a bijective, continuous homomorphism, but it will not necessarily be a homeomorphism.\nIn other words, it will not necessarily admit an inverse in the [category](https://bliptext.com/articles/category-theory) of topological groups.\nThere 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\nG/ker(f)\nto\nim(f)\nis an isomorphism if and only if the map f is open onto its image.\nThe third isomorphism theorem, however, is true more or less verbatim for topological groups, as one may easily check.\n\n# Hilbert's fifth problem\nThere are several strong results on the relation between topological groups and Lie groups.\nFirst, every continuous homomorphism of Lie groups G \\to H is smooth.\nIt follows that a topological group has a unique structure of a Lie group if one exists.\nAlso, [Cartan's theorem](https://bliptext.com/articles/closed-subgroup-theorem) says that every closed subgroup of a Lie group is a Lie subgroup, in particular a smooth [submanifold](https://bliptext.com/articles/submanifold).\n**[Hilbert's fifth problem](https://bliptext.com/articles/hilbert-s-fifth-problem)** asked whether a topological group G that is a [topological manifold](https://bliptext.com/articles/topological-manifold) must be a Lie group.\nIn other words, does G have the structure of a smooth manifold, making the group operations smooth?\nAs shown by [Andrew Gleason](https://bliptext.com/articles/andrew-gleason), [Deane Montgomery](https://bliptext.com/articles/deane-montgomery), and [Leo Zippin](https://bliptext.com/articles/leo-zippin), the answer to this problem is yes.\nIn fact, G has a [real analytic](https://bliptext.com/articles/real-analytic) structure.\nUsing the smooth structure, one can define the Lie algebra of G, an object of [linear algebra](https://bliptext.com/articles/linear-algebra) that determines a [connected](https://bliptext.com/articles/connected-space) group G up to [covering spaces](https://bliptext.com/articles/covering-space).\nAs 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.\nThe theorem also has consequences for broader classes of topological groups.\nFirst, every [compact group](https://bliptext.com/articles/compact-group) (understood to be Hausdorff) is an inverse limit of compact Lie groups.\n(One important case is an inverse limit of finite groups, called a [profinite group](https://bliptext.com/articles/profinite-group). For example, the group\n\\mathbb{Z}p\nof p-adic integers and the [absolute Galois group](https://bliptext.com/articles/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\nGL(n,\\mathbb{Q}p)\ncontains the compact open subgroup\nGL(n,\\mathbb{Z}p)\n, which is the inverse limit of the finite groups\nGL(n,\\mathbb{Z}/pr)\nas r' goes to infinity.)\n\n# Representations of compact or locally compact groups\nAn **action** of a topological group G on a topological space X is a group **action** of G on X such that the corresponding function\nG × X → X\nis continuous.\nLikewise, a **[representation](https://bliptext.com/articles/representation-theory)** of a topological group G on a real or complex topological vector space V is a continuous action of G on V such that for each\ng ∈ G\n, the map\nv ↦ gv\nfrom V to itself is linear.\nGroup actions and representation theory are particularly well understood for compact groups, generalizing what happens for [finite groups](https://bliptext.com/articles/representation-theory-of-finite-groups).\nFor example, every finite-dimensional (real or complex) representation of a compact group is a [direct sum](https://bliptext.com/articles/direct-sum) of [irreducible representations](https://bliptext.com/articles/irreducible-representation).\nAn infinite-dimensional [unitary representation](https://bliptext.com/articles/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](https://bliptext.com/articles/peter-weyl-theorem).\nFor example, the theory of [Fourier series](https://bliptext.com/articles/fourier-series) describes the decomposition of the unitary representation of the circle group\nS1\non the complex Hilbert space\nL2(S1)\n. The irreducible representations of\nS1\nare all 1-dimensional, of the form\nz ↦ zn\nfor integers n (where\nS1\nis viewed as a subgroup of the multiplicative group \\mathbb{C}*). Each of these representations occurs with multiplicity 1 in\nL2(S1)\n.\nThe irreducible representations of all compact connected Lie groups have been classified.\nIn particular, the [character](https://bliptext.com/articles/character-theory) of each irreducible representation is given by the Weyl [character](https://bliptext.com/articles/character-theory) formula.\nMore generally, locally compact groups have a rich theory of [harmonic analysis](https://bliptext.com/articles/harmonic-analysis), because they admit a natural notion of [measure](https://bliptext.com/articles/measure-mathematics) and [integral](https://bliptext.com/articles/integral), given by the Haar [measure](https://bliptext.com/articles/measure-mathematics).\nEvery unitary representation of a locally compact group can be described as a [direct integral](https://bliptext.com/articles/direct-integral) of irreducible unitary representations.\n(The decomposition is essentially unique if G is of [Type I](https://bliptext.com/articles/c-algebra), which includes the most important examples such as abelian groups and [semisimple Lie groups](https://bliptext.com/articles/semisimple-lie-group).)\nA basic example is the [Fourier transform](https://bliptext.com/articles/fourier-transform), which decomposes the action of the additive group \\mathbb{R} on the Hilbert space\nL2(\\mathbb{R})\nas a direct integral of the irreducible unitary representations of \\mathbb{R}.\nThe irreducible unitary representations of \\mathbb{R} are all 1-dimensional, of the form\nx ↦ e2πiax\nfor\na ∈ \\mathbb{R}\n.\nThe irreducible unitary representations of a locally compact group may be infinite-dimensional.\nA major goal of representation theory, related to the [Langlands classification](https://bliptext.com/articles/langlands-classification) of [admissible representations](https://bliptext.com/articles/admissible-representation), is to find the [unitary dual](https://bliptext.com/articles/unitary-dual) (the space of all irreducible unitary representations) for the semisimple Lie groups.\nThe unitary dual is known in many cases such as\nSL(2,\\mathbb{R})\n, but not all.\nFor a [locally compact abelian group](https://bliptext.com/articles/locally-compact-abelian-group) G, every irreducible unitary representation has dimension 1.\nIn this case, the unitary dual \\hat{G} is a group, in fact another locally compact abelian group.\n[Pontryagin duality](https://bliptext.com/articles/pontryagin-duality) states that for a locally compact abelian group G, the dual of \\hat{G} is the original group G. For example, the dual group of the integers\n\\mathbb{Z}\nis the circle group\nS1\n, while the group \\mathbb{R} of real numbers is isomorphic to its own dual.\nEvery locally compact group G has a good supply of irreducible unitary representations; for example, enough representations to distinguish the points of G (the [Gelfand–Raikov theorem](https://bliptext.com/articles/gelfand-raikov-theorem)).\nBy 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.\nFor example, there are many abelian [Banach–Lie groups](https://bliptext.com/articles/banach-manifold) for which every representation on Hilbert space is trivial.\n\n# Homotopy theory of topological groups\nTopological groups are special among all topological spaces, even in terms of their [homotopy type](https://bliptext.com/articles/homotopy-type).\nOne basic point is that a topological group G determines a path-connected topological space, the [classifying space](https://bliptext.com/articles/classifying-space)\nBG\n(which classifies [principal G-bundles](https://bliptext.com/articles/principal-bundle) over topological spaces, under mild hypotheses).\nThe group G is isomorphic in the [homotopy category](https://bliptext.com/articles/model-category) to the [loop space](https://bliptext.com/articles/loop-space) of\nBG\nSome of these restrictions hold in the broader context of [H-spaces](https://bliptext.com/articles/h-space).\nFor example, the [fundamental group](https://bliptext.com/articles/fundamental-group) of a topological group G is abelian.\n(More generally, the [Whitehead product](https://bliptext.com/articles/whitehead-product) on the homotopy groups of G is zero.)\nAlso, for any field k, the [cohomology](https://bliptext.com/articles/cohomology) ring\nH*(G,k)\nhas the structure of a [Hopf algebra](https://bliptext.com/articles/hopf-algebra).\nIn view of structure theorems on Hopf algebras by [Heinz Hopf](https://bliptext.com/articles/heinz-hopf) and [Armand Borel](https://bliptext.com/articles/armand-borel), this puts strong restrictions on the possible cohomology rings of topological groups.\nIn particular, if G is a path-connected topological group whose rational cohomology ring\nH*(G,\\mathbb{Q})\nis finite-dimensional in each degree, then this ring must be a free [graded-commutative](https://bliptext.com/articles/graded-commutative) algebra over \\mathbb{Q}, that is, the [tensor product](https://bliptext.com/articles/tensor-product-of-algebras) of a [polynomial ring](https://bliptext.com/articles/polynomial-ring) on generators of even degree with an [exterior algebra](https://bliptext.com/articles/exterior-algebra) on generators of odd degree.\nIn particular, for a connected Lie group G, the rational cohomology ring of G is an exterior algebra on generators of odd degree.\nMoreover, a connected Lie group G has a [maximal compact subgroup](https://bliptext.com/articles/maximal-compact-subgroup) K, which is unique up to conjugation, and the inclusion of K into G is a [homotopy equivalence](https://bliptext.com/articles/homotopy-equivalence).\nSo describing the homotopy types of Lie groups reduces to the case of compact Lie groups.\nFor example, the maximal compact subgroup of\nSL(2,\\mathbb{R})\nis the circle group\nSO(2)\n, and the homogeneous space\nSL(2,\\mathbb{R})/SO(2)\ncan be identified with the [hyperbolic plane](https://bliptext.com/articles/hyperbolic-plane).\nSince the hyperbolic plane is [contractible](https://bliptext.com/articles/contractible-space), the inclusion of the circle group into\nSL(2,\\mathbb{R})\nis a homotopy equivalence.\nFinally, compact connected Lie groups have been classified by [Wilhelm Killing](https://bliptext.com/articles/wilhelm-killing), [Élie Cartan](https://bliptext.com/articles/lie-cartan), and [Hermann Weyl](https://bliptext.com/articles/hermann-weyl).\nAs a result, there is an essentially complete description of the possible homotopy types of Lie groups.\nFor example, a compact connected Lie group of dimension at most 3 is either a torus, the group [SU(2)](https://bliptext.com/articles/su-2) ([diffeomorphic](https://bliptext.com/articles/diffeomorphic) to the 3-sphere\nS3\n), or its quotient group\nSU(2)/{±1} ≅ SO(3)\n(diffeomorphic to\nRP3\n).\n\n# Complete topological group\nInformation about convergence of nets and filters, such as definitions and properties, can be found in the article about [filters in topology](https://bliptext.com/articles/filters-in-topology).\n\n## Canonical uniformity on a commutative topological group\nThis article will henceforth assume that any topological group that we consider is an additive commutative topological group with identity element 0.\nThe **diagonal** of X is the set\nand for any containing 0, the **canonical entourage** or **canonical vicinities around N** is the set\nFor a topological group (X, \\tau), the **canonical uniformity** on X is the [uniform structure](https://bliptext.com/articles/uniform-space) induced by the set of all canonical entourages \\Delta(N) as N ranges over all neighborhoods of 0 in X.\nThat is, it is the upward closure of the following prefilter on X \\times X,\nwhere this prefilter forms what is known as a [base of entourages](https://bliptext.com/articles/base-of-entourages) of the canonical uniformity.\nFor a commutative additive group X, a fundamental system of entourages \\mathcal{B} is called a ****translation-invariant** uniformity** if for every if and only if for all A uniformity \\mathcal{B} is called **translation-invariant** if it has a base of entourages that is **translation-invariant**.\n

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(N) contains the diagonal because 0 \\in N.
If N is [symmetric](https://bliptext.com/articles/symmetric-set) (that is, -N = N) then \\Delta_X(N) is [symmetric](https://bliptext.com/articles/symmetric-set) (meaning that ) and\n
The topology induced on X by the canonical uniformity is the same as the topology that X started with (that is, it is \\tau).

\n\n## Cauchy prefilters and nets\nThe general theory of [uniform spaces](https://bliptext.com/articles/uniform-space) has its own definition of a \"Cauchy prefilter\" and \"Cauchy net.\"\nFor the canonical uniformity on X, these reduces down to the definition described below.\nSuppose is a net in X and is a net in Y. Make I \\times J into a directed set by declaring if and only if Then denotes the **product net**.\nIf X = Y then the image of this net under the addition map denotes the **sum** of these two nets:\nand similarly their **difference** is defined to be the image of the product net under the subtraction map:\nA [net](https://bliptext.com/articles/net-mathematics) in an additive topological group X is called a **Cauchy [net](https://bliptext.com/articles/net-mathematics)** if\nor equivalently, if for every neighborhood N of 0 in X, there exists some i_0 \\in I such that\nfor all indices\nA **[Cauchy sequence](https://bliptext.com/articles/cauchy-sequence)** is a Cauchy net that is a sequence.\nIf B is a subset of an additive group X and N is a set containing 0, thenB is said to be an **N-small set** or **small of order N** if\nA prefilter \\mathcal{B} on an additive topological group X called a **Cauchy prefilter** if it satisfies any of the following equivalent conditions:\n

in X, where is a prefilter.
in X, where is a prefilter equivalent to
For every neighborhood N of 0 in X, \\mathcal{B} contains some N-small set (that is, there exists some such that ).

\nand if X is commutative then also:\n

For every neighborhood N of 0 in X, there exists some and some x \\in X such that

\n\nSuppose \\mathcal{B} is a prefilter on a commutative topological group X and x \\in X. Then in X if and only if and \\mathcal{B} is Cauchy.\n\n## Complete commutative topological group\nRecall that for any a prefilter \\mathcal{C} **on S** is necessarily a subset of \\wp(S); that is,\nA subset S of a topological group X is called a **complete subset** if it satisfies any of the following equivalent conditions:\n

Every Cauchy prefilter on S [converges](https://bliptext.com/articles/filters-in-topology) to at least one point of S.
Every Cauchy net in S [converges](https://bliptext.com/articles/filters-in-topology) to at least one point of S;
Every Cauchy filter \\mathcal{C} on S converges to at least one point of S.
S is a [complete](https://bliptext.com/articles/complete-uniform-space) uniform space (under the point-set topology definition of \"[complete](https://bliptext.com/articles/complete-uniform-space) uniform space\") when S is endowed with the uniformity induced on it by the canonical uniformity of X;

\nA 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.\nA commutative topological group X is called a **complete group** if any of the following equivalent conditions hold:\n

X is complete as a subset of itself.
Every Cauchy net in X [converges](https://bliptext.com/articles/net-mathematics) to at least one point of X.
There exists a neighborhood of 0 in X that is also a complete subset of X.
When endowed with its canonical uniformity, X becomes is a [complete uniform space](https://bliptext.com/articles/complete-uniform-space).\n

\nA topological group is called **sequentially complete** if it is a **sequentially complete** subset of itself.\n**Neighborhood basis**: Suppose C is a completion of a commutative topological group X with and that \\mathcal{N} is a [neighborhood base](https://bliptext.com/articles/neighborhood-base) of the origin in X. Then the family of sets\nis a neighborhood basis at the origin in C.\n****\nLet X and Y be topological groups, and f : D \\to Y be a map.\nThen 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\n\n# Generalizations\nVarious generalizations of topological groups can be obtained by weakening the continuity conditions:\nc ∈ G\nthe two functions\nG → G\ndefined by\nx ↦ xc\nand\nx ↦ cx\nare continuous.\n\n# Citations","current_revision":null,"created_by":"kACnuQmH5k9Nr5VNVEpVeg7XpvUmMctD","created_at":"2025-01-25T17:33:35.965Z","updated_at":"2025-01-25T17:33:35.965Z","search_vector":"'group':2 'topolog':1","search_content":" \n \n\n# \nIn [mathematics](https://bliptext.com/articles/mathematics), **topological [groups](https://bliptext.com/articles/group-mathematics)** are the combination of [groups](https://bliptext.com/articles/group-mathematics) and [topological spaces](https://bliptext.com/articles/topological-space), i.e. they are [groups](https://bliptext.com/articles/group-mathematics) and [topological spaces](https://bliptext.com/articles/topological-space) at the same time, such that the [continuity](https://bliptext.com/articles/continuous-function) condition for the group operations connects these two structures together and consequently they are not independent from each other.\nTopological groups have been studied extensively in the period of 1925 to 1940.\n[Haar](https://bliptext.com/articles/alfr-d-haar) and [Weil](https://bliptext.com/articles/andr-weil) (respectively in 1933 and 1940) showed that the [integrals](https://bliptext.com/articles/integral) and [Fourier series](https://bliptext.com/articles/fourier-series) are special cases of a very wide class of topological groups.\nTopological groups, along with [continuous group actions](https://bliptext.com/articles/continuous-group-action), are used to study continuous [symmetries](https://bliptext.com/articles/symmetry), which have many applications, for example, [in physics](https://bliptext.com/articles/symmetry-physics).\nIn [functional analysis](https://bliptext.com/articles/functional-analysis), every [topological vector space](https://bliptext.com/articles/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](https://bliptext.com/articles/functional-analysis).\n# [The real numbers form a topological group under addition | upload.wikimedia.org/wikipedia/commons/d/d7/Real///number///line.svg]\n\n# Formal definition\nA **topological group**, G, is a [topological space](https://bliptext.com/articles/topological-space) that is also a group such that the group operation (in this case product):\n⋅ : G × G → G\n,\n(x, y) ↦ xy\nand the inversion map:\n−1 : G → G\n,\nx ↦ x−1\nare [continuous](https://bliptext.com/articles/continuous-function).\nHere\nG × G\nis viewed as a topological space with the [product topology](https://bliptext.com/articles/product-topology).\nSuch a topology is said to be **compatible with the group operations** and is called a **group topology**.\nThe product map is continuous if and only if for any\nx, y ∈ G\nand any neighborhood W of\nxy\nin G, there exist neighborhoods U of x and V of y in G such that\nU ⋅ V ⊆ W\n, where\nU ⋅ V := {u ⋅ v : u ∈ U, v ∈ V}\n. The inversion map is continuous if and only if for any\nx ∈ G\nand any neighborhood V of\nx−1\nin G, there exists a neighborhood U of x in G such that\nU−1 ⊆ V\n, where\nU−1 := { u−1 : u ∈ U }\n.\nTo show that a topology is compatible with the group operations, it suffices to check that the map\nG × G → G\n,\n(x, y) ↦ xy−1\nis continuous.\nExplicitly, this means that for any\nx, y ∈ G\nand any neighborhood W in G of\nxy−1\n, there exist neighborhoods U of x and V of y in G such that\nU ⋅ (V−1) ⊆ W\n.\nThis definition used notation for multiplicative groups;\nthe equivalent for additive groups would be that the following two operations are continuous:\n+ : G × G → G\n,\n(x, y) ↦ x + y\n− : G → G\n,\nx ↦ −x\n.\nAlthough not part of this definition, many authors require that the topology on G be [Hausdorff](https://bliptext.com/articles/hausdorff-space).\nOne reason for this is that any topological group can be canonically associated with a Hausdorff topological group by taking an appropriate canonical quotient;\nthis however, often still requires working with the original non-Hausdorff topological group.\nOther reasons, and some equivalent conditions, are discussed below.\nThis article will not assume that topological groups are necessarily Hausdorff.\nIn the language of [category theory](https://bliptext.com/articles/category-theory), topological groups can be defined concisely as [group objects](https://bliptext.com/articles/group-object) in the [category of topological spaces](https://bliptext.com/articles/category-of-topological-spaces), in the same way that ordinary groups are [group objects](https://bliptext.com/articles/group-object) in the [category of sets](https://bliptext.com/articles/category-of-sets).\nNote that the axioms are given in terms of the maps (binary product, unary inverse, and nullary identity), hence are categorical definitions.\n\n## Homomorphisms\nA **homomorphism** of topological groups means a continuous group **homomorphism**\nG → H\n. Topological groups, together with their homomorphisms, form a [category](https://bliptext.com/articles/category-theory).\nA group homomorphism between topological groups is continuous if and only if it is continuous at some point.\nAn **isomorphism** of topological groups is a group **isomorphism** that is also a [homeomorphism](https://bliptext.com/articles/homeomorphism) of the underlying topological spaces.\nThis is stronger than simply requiring a continuous group isomorphism—the inverse must also be continuous.\nThere are examples of topological groups that are isomorphic as ordinary groups but not as topological groups.\nIndeed, any non-discrete topological group is also a topological group when considered with the discrete topology.\nThe underlying groups are the same, but as topological groups there is not an isomorphism.\n\n# Examples\nEvery group can be trivially made into a topological group by considering it with the [discrete topology](https://bliptext.com/articles/discrete-topology); such groups are called [discrete groups](https://bliptext.com/articles/discrete-group).\nIn this sense, the theory of topological groups subsumes that of ordinary groups.\nThe [indiscrete topology](https://bliptext.com/articles/indiscrete-topology) (i.e. the trivial topology) also makes every group into a topological group.\nThe [real numbers](https://bliptext.com/articles/real-number), \\mathbb{R} with the usual topology form a topological group under addition.\n[Euclidean n-space](https://bliptext.com/articles/euclidean-space)\n\\mathbb{R}n\nis also a topological group under addition, and more generally, every [topological vector space](https://bliptext.com/articles/topological-vector-space) forms an (abelian) topological group.\nSome other examples of [abelian](https://bliptext.com/articles/abelian-group) topological groups are the [circle group](https://bliptext.com/articles/circle-group)\nS1\n, or the [torus](https://bliptext.com/articles/torus-group)\n(S1)n\nfor any natural number n.\nThe [classical groups](https://bliptext.com/articles/classical-group) are important examples of non-abelian topological groups.\nFor instance, the [general linear group](https://bliptext.com/articles/general-linear-group)\nGL(n,\\mathbb{R})\nof all invertible n-by-n [matrices](https://bliptext.com/articles/matrix-mathematics) with real entries can be viewed as a topological group with the topology defined by viewing\nGL(n,\\mathbb{R})\nas a [subspace](https://bliptext.com/articles/subspace-topology) of Euclidean space\n\\mathbb{R}n×n\n. Another classical group is the [orthogonal group](https://bliptext.com/articles/orthogonal-group)\nO(n)\n, the group of all [linear maps](https://bliptext.com/articles/linear-map) from\n\\mathbb{R}n\nto itself that preserve the [length](https://bliptext.com/articles/euclidean-distance) of all vectors.\nThe orthogonal group is [compact](https://bliptext.com/articles/compact-space) as a topological space.\nMuch of [Euclidean geometry](https://bliptext.com/articles/euclidean-geometry) can be viewed as studying the structure of the orthogonal group, or the closely related group\nO(n) ⋉ \\mathbb{R}n\nof [isometries](https://bliptext.com/articles/euclidean-group) of\n\\mathbb{R}n\n.\nThe groups mentioned so far are all [Lie groups](https://bliptext.com/articles/lie-group), meaning that they are [[smooth](https://bliptext.com/articles/smooth-function) manifolds](https://bliptext.com/articles/[smooth](https://bliptext.com/articles/smooth-function)-manifold) in such a way that the group operations are [smooth](https://bliptext.com/articles/smooth-function), not just continuous.\nLie groups are the best-understood topological groups; many questions about Lie groups can be converted to purely algebraic questions about [Lie algebras](https://bliptext.com/articles/lie-algebra) and then solved.\nAn example of a topological group that is not a Lie group is the additive group \\mathbb{Q} of [rational numbers](https://bliptext.com/articles/rational-number), with the topology inherited from \\mathbb{R}.\nThis is a [countable](https://bliptext.com/articles/countable) space, and it does not have the discrete topology.\nAn important example for [number theory](https://bliptext.com/articles/number-theory) is the group\n\\mathbb{Z}p\nof p-adic integers, for a [prime number](https://bliptext.com/articles/prime-number) p, meaning the [inverse limit](https://bliptext.com/articles/inverse-limit) of the finite groups\n\\mathbb{Z}/pn\nas n goes to infinity.\nThe group\n\\mathbb{Z}p\nis well behaved in that it is compact (in fact, homeomorphic to the [Cantor set](https://bliptext.com/articles/cantor-set)), but it differs from (real) Lie groups in that it is [totally disconnected](https://bliptext.com/articles/totally-disconnected-group).\nMore generally, there is a theory of p-adic Lie groups, including compact groups such as\nGL(n,\\mathbb{Z}p)\nas well as [locally compact groups](https://bliptext.com/articles/locally-compact-group) such as\nGL(n,\\mathbb{Q}p)\n, where\n\\mathbb{Q}p\nis the locally compact [field](https://bliptext.com/articles/field-mathematics) of p-adic numbers.\nThe group\n\\mathbb{Z}p\nis a [pro-finite group](https://bliptext.com/articles/pro-finite-group); it is isomorphic to a subgroup of the product in such a way that its topology is induced by the product topology, where the finite groups are given the discrete topology.\nAnother large class of pro-finite groups important in number theory are [absolute Galois groups](https://bliptext.com/articles/absolute-galois-group).\nSome topological groups can be viewed as [infinite dimensional Lie groups](https://bliptext.com/articles/infinite-dimensional-lie-group); this phrase is best understood informally, to include several different families of examples.\nFor example, a [topological vector space](https://bliptext.com/articles/topological-vector-space), such as a [Banach space](https://bliptext.com/articles/banach-space) or [Hilbert space](https://bliptext.com/articles/hilbert-space), is an abelian topological group under addition.\nSome other infinite-dimensional groups that have been studied, with varying degrees of success, are [loop groups](https://bliptext.com/articles/loop-group), [Kac–Moody groups](https://bliptext.com/articles/kac-moody-algebra), [Diffeomorphism groups](https://bliptext.com/articles/diffeomorphism), [homeomorphism groups](https://bliptext.com/articles/homeomorphism-group), and [gauge groups](https://bliptext.com/articles/gauge-group).\nIn every [Banach algebra](https://bliptext.com/articles/banach-algebra) with multiplicative identity, the set of invertible elements forms a topological group under multiplication.\nFor example, the group of invertible [bounded operators](https://bliptext.com/articles/bounded-operator) on a Hilbert space arises this way.\n\n# Properties\n\n## Translation invariance\nEvery 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 the subset S is [open](https://bliptext.com/articles/open-set) (resp. [closed](https://bliptext.com/articles/closed-set)) in G if and only if this is true of its left translation and right translation\nIf \\mathcal{N} is a [neighborhood basis](https://bliptext.com/articles/neighborhood-basis) of the identity element in a topological group G then for all x \\in X,\nis 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.\nIf S is any subset of G and U is an open subset of G, then is an open subset of G.\n\n## Symmetric neighborhoods\nThe inversion operation on a topological group G is a homeomorphism from G to itself.\nA subset is said to be [symmetric](https://bliptext.com/articles/symmetric-set) if S^{-1} = S, where\nThe closure of every symmetric set in a commutative topological group is symmetric.\nIf S is any subset of a commutative topological group G, then the following sets are also symmetric:\nS−1 ∩ S\n,\nS−1 ∪ S\n, and\nS−1 S\n.\nFor any neighborhood N in a commutative topological group G of the identity element, there exists a symmetric neighborhood M of the identity element such that\nM−1 M ⊆ N\n, where note that\nM−1 M\nis necessarily a symmetric neighborhood of the identity element.\nThus every topological group has a neighborhood basis at the identity element consisting of symmetric sets.\nIf G is a [locally compact](https://bliptext.com/articles/locally-compact) commutative group, then for any neighborhood N in G of the identity element, there exists a symmetric relatively compact neighborhood M of the identity element such that\ncl M ⊆ N\n(where\ncl M\nis symmetric as well).\n\n## Uniform space\nEvery topological group can be viewed as a [uniform space](https://bliptext.com/articles/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.\nIf G is not abelian, then these two need not coincide.\nThe uniform structures allow one to talk about notions such as [completeness](https://bliptext.com/articles/completeness-topology), [uniform continuity](https://bliptext.com/articles/uniformly-continuous) and [uniform convergence](https://bliptext.com/articles/uniform-convergence) on topological groups.\n\n## Separation properties\nIf U is an open subset of a commutative topological group G and U contains a compact set K, then there exists a neighborhood N of the identity element such that\nKN ⊆ U\n.\nAs a uniform space, every commutative topological group is [completely regular](https://bliptext.com/articles/completely-regular-space).\nConsequently, for a multiplicative topological group G with identity element 1, the following are equivalent:\n \n G is a T0-space ([Kolmogorov](https://bliptext.com/articles/kolmogorov-space)); \n G is a T2-space ([Hausdorff](https://bliptext.com/articles/hausdorff-space)); \n G is a T31/2 ([Tychonoff](https://bliptext.com/articles/tychonoff-space)); \n \n{ 1 }\nis closed in G; \n \n{ 1 } := ∩ N ∈ 𝒩 N\n, where\n𝒩\nis a neighborhood basis of the identity element in G; \n for any x \\in G such that x \\neq 1, there exists a neighborhood U in G of the identity element such that \n \nA subgroup of a commutative topological group is discrete if and only if it has an [isolated point](https://bliptext.com/articles/isolated-point).\nIf G is not Hausdorff, then one can obtain a Hausdorff group by passing to the quotient group\nG/K\n, where K is the [closure](https://bliptext.com/articles/closure-topology) of the identity.\nThis is equivalent to taking the [Kolmogorov quotient](https://bliptext.com/articles/kolmogorov-space) of G.\n\n## Metrisability\nLet G be a topological group.\nAs with any topological space, we say that G is [metrisable](https://bliptext.com/articles/metrizable-space) if and only if there exists a metric d on G, which induces the same topology on G. A metric d on G is called\nThe **Birkhoff–Kakutani theorem** (named after mathematicians [Garrett Birkhoff](https://bliptext.com/articles/garrett-birkhoff) and [Shizuo Kakutani](https://bliptext.com/articles/shizuo-kakutani)) states that the following three conditions on a topological group G are equivalent:\nFurthermore, the following are equivalent for any topological group G:\n**Note:** As with the rest of the article we of assume here a Hausdorff topology.\nThe implications 4 \\Rightarrow 3 \\Rightarrow 2 \\Rightarrow 1 hold in any topological space.\nIn 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](https://bliptext.com/articles/compact-space)) subsets.\nThe non-trivial implication 1 \\Rightarrow 4 was first proved by Raimond Struble in 1974.\nAn alternative approach was made by [Uffe Haagerup](https://bliptext.com/articles/uffe-haagerup) and Agata Przybyszewska in 2006,\nthe idea of the which is as follows:\nOne relies on the construction of a left-invariant metric, d_{0}, as in the case of [first countable spaces](https://bliptext.com/articles/first-countable-space).\nBy local compactness, closed balls of sufficiently small radii are compact, and by normalising we can assume this holds for radius 1.\nClosing the open ball, U, of radius 1 under multiplication yields a [clopen](https://bliptext.com/articles/clopen) subgroup, H, of G, on which the metric d_{0} is proper.\nSince H is open and G is [second countable](https://bliptext.com/articles/second-countable-space), the subgroup has at most countably many cosets.\nOne now uses this sequence of cosets and the metric on H to construct a proper metric on G.\n\n## Subgroups\nEvery [subgroup](https://bliptext.com/articles/subgroup) of a topological group is itself a topological group when given the [subspace topology](https://bliptext.com/articles/subspace-topology).\nEvery open subgroup H is also closed in G, since the complement of H is the open set given by the union of open sets\ngH\nfor\ng ∈ G \\ H\n. If H is a subgroup of G then the closure of H is also a subgroup.\nLikewise, if H is a normal subgroup of G, the closure of H is normal in G.\n\n## Quotients and normal subgroups\nIf H is a subgroup of G, the set of left [cosets](https://bliptext.com/articles/coset)\nG/H\nwith the [quotient topology](https://bliptext.com/articles/quotient-topology) is called a [homogeneous space](https://bliptext.com/articles/homogeneous-space) for G. The quotient map is always [open](https://bliptext.com/articles/open-map).\nFor example, for a positive integer n, the [sphere](https://bliptext.com/articles/n-sphere)\nSn\nis a homogeneous space for the [rotation group](https://bliptext.com/articles/rotation-group)\nSO(n+1)\nin\n\\Rn+1\n, with\n. A homogeneous space\nG/H\nis Hausdorff if and only if H is closed in G. Partly for this reason, it is natural to concentrate on closed subgroups when studying topological groups.\nIf H is a [normal subgroup](https://bliptext.com/articles/normal-subgroup) of G, then the [quotient group](https://bliptext.com/articles/quotient-group)\nG/H\nbecomes a topological group when given the quotient topology.\nIt is Hausdorff if and only if H is closed in G. For example, the quotient group\n\\R / \\Z\nis isomorphic to the circle group\nS1\n.\nIn any topological group, the [identity component](https://bliptext.com/articles/identity-component) (i.e., the [connected component](https://bliptext.com/articles/connected-component-topology) containing the identity element) is a closed normal subgroup.\nIf C is the identity component and a is any point of G, then the left coset\naC\nis the component of G containing a.\nSo the collection of all left cosets (or right cosets) of C in G is equal to the collection of all components of G. It follows that the quotient group\nG/C\nis [totally disconnected](https://bliptext.com/articles/totally-disconnected).\n\n## Closure and compactness\nIn any commutative topological group, the product (assuming the group is multiplicative)\nKC\nof a compact set K and a closed set C is a closed set.\nFurthermore, for any subsets R and S of G,\n(cl R)(cl S) ⊆ cl (RS)\n.\nIf H is a subgroup of a commutative topological group G and if N is a neighborhood in G of the identity element such that H ∩ cl N is closed, then H is closed.\nEvery discrete subgroup of a Hausdorff commutative topological group is closed.\n\n## Isomorphism theorems\nThe [isomorphism theorems](https://bliptext.com/articles/isomorphism-theorem) from ordinary group theory are not always true in the topological setting.\nThis is because a bijective homomorphism need not be an isomorphism of topological groups.\nFor 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 is an isomorphism of topological groups; it will be a bijective, continuous homomorphism, but it will not necessarily be a homeomorphism.\nIn other words, it will not necessarily admit an inverse in the [category](https://bliptext.com/articles/category-theory) of topological groups.\nThere 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\nG/ker(f)\nto\nim(f)\nis an isomorphism if and only if the map f is open onto its image.\nThe third isomorphism theorem, however, is true more or less verbatim for topological groups, as one may easily check.\n\n# Hilbert's fifth problem\nThere are several strong results on the relation between topological groups and Lie groups.\nFirst, every continuous homomorphism of Lie groups G \\to H is smooth.\nIt follows that a topological group has a unique structure of a Lie group if one exists.\nAlso, [Cartan's theorem](https://bliptext.com/articles/closed-subgroup-theorem) says that every closed subgroup of a Lie group is a Lie subgroup, in particular a smooth [submanifold](https://bliptext.com/articles/submanifold).\n**[Hilbert's fifth problem](https://bliptext.com/articles/hilbert-s-fifth-problem)** asked whether a topological group G that is a [topological manifold](https://bliptext.com/articles/topological-manifold) must be a Lie group.\nIn other words, does G have the structure of a smooth manifold, making the group operations smooth?\nAs shown by [Andrew Gleason](https://bliptext.com/articles/andrew-gleason), [Deane Montgomery](https://bliptext.com/articles/deane-montgomery), and [Leo Zippin](https://bliptext.com/articles/leo-zippin), the answer to this problem is yes.\nIn fact, G has a [real analytic](https://bliptext.com/articles/real-analytic) structure.\nUsing the smooth structure, one can define the Lie algebra of G, an object of [linear algebra](https://bliptext.com/articles/linear-algebra) that determines a [connected](https://bliptext.com/articles/connected-space) group G up to [covering spaces](https://bliptext.com/articles/covering-space).\nAs 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.\nThe theorem also has consequences for broader classes of topological groups.\nFirst, every [compact group](https://bliptext.com/articles/compact-group) (understood to be Hausdorff) is an inverse limit of compact Lie groups.\n(One important case is an inverse limit of finite groups, called a [profinite group](https://bliptext.com/articles/profinite-group). For example, the group\n\\mathbb{Z}p\nof p-adic integers and the [absolute Galois group](https://bliptext.com/articles/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\nGL(n,\\mathbb{Q}p)\ncontains the compact open subgroup\nGL(n,\\mathbb{Z}p)\n, which is the inverse limit of the finite groups\nGL(n,\\mathbb{Z}/pr)\nas r' goes to infinity.)\n\n# Representations of compact or locally compact groups\nAn **action** of a topological group G on a topological space X is a group **action** of G on X such that the corresponding function\nG × X → X\nis continuous.\nLikewise, a **[representation](https://bliptext.com/articles/representation-theory)** of a topological group G on a real or complex topological vector space V is a continuous action of G on V such that for each\ng ∈ G\n, the map\nv ↦ gv\nfrom V to itself is linear.\nGroup actions and representation theory are particularly well understood for compact groups, generalizing what happens for [finite groups](https://bliptext.com/articles/representation-theory-of-finite-groups).\nFor example, every finite-dimensional (real or complex) representation of a compact group is a [direct sum](https://bliptext.com/articles/direct-sum) of [irreducible representations](https://bliptext.com/articles/irreducible-representation).\nAn infinite-dimensional [unitary representation](https://bliptext.com/articles/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](https://bliptext.com/articles/peter-weyl-theorem).\nFor example, the theory of [Fourier series](https://bliptext.com/articles/fourier-series) describes the decomposition of the unitary representation of the circle group\nS1\non the complex Hilbert space\nL2(S1)\n. The irreducible representations of\nS1\nare all 1-dimensional, of the form\nz ↦ zn\nfor integers n (where\nS1\nis viewed as a subgroup of the multiplicative group \\mathbb{C}*). Each of these representations occurs with multiplicity 1 in\nL2(S1)\n.\nThe irreducible representations of all compact connected Lie groups have been classified.\nIn particular, the [character](https://bliptext.com/articles/character-theory) of each irreducible representation is given by the Weyl [character](https://bliptext.com/articles/character-theory) formula.\nMore generally, locally compact groups have a rich theory of [harmonic analysis](https://bliptext.com/articles/harmonic-analysis), because they admit a natural notion of [measure](https://bliptext.com/articles/measure-mathematics) and [integral](https://bliptext.com/articles/integral), given by the Haar [measure](https://bliptext.com/articles/measure-mathematics).\nEvery unitary representation of a locally compact group can be described as a [direct integral](https://bliptext.com/articles/direct-integral) of irreducible unitary representations.\n(The decomposition is essentially unique if G is of [Type I](https://bliptext.com/articles/c-algebra), which includes the most important examples such as abelian groups and [semisimple Lie groups](https://bliptext.com/articles/semisimple-lie-group).)\nA basic example is the [Fourier transform](https://bliptext.com/articles/fourier-transform), which decomposes the action of the additive group \\mathbb{R} on the Hilbert space\nL2(\\mathbb{R})\nas a direct integral of the irreducible unitary representations of \\mathbb{R}.\nThe irreducible unitary representations of \\mathbb{R} are all 1-dimensional, of the form\nx ↦ e2πiax\nfor\na ∈ \\mathbb{R}\n.\nThe irreducible unitary representations of a locally compact group may be infinite-dimensional.\nA major goal of representation theory, related to the [Langlands classification](https://bliptext.com/articles/langlands-classification) of [admissible representations](https://bliptext.com/articles/admissible-representation), is to find the [unitary dual](https://bliptext.com/articles/unitary-dual) (the space of all irreducible unitary representations) for the semisimple Lie groups.\nThe unitary dual is known in many cases such as\nSL(2,\\mathbb{R})\n, but not all.\nFor a [locally compact abelian group](https://bliptext.com/articles/locally-compact-abelian-group) G, every irreducible unitary representation has dimension 1.\nIn this case, the unitary dual \\hat{G} is a group, in fact another locally compact abelian group.\n[Pontryagin duality](https://bliptext.com/articles/pontryagin-duality) states that for a locally compact abelian group G, the dual of \\hat{G} is the original group G. For example, the dual group of the integers\n\\mathbb{Z}\nis the circle group\nS1\n, while the group \\mathbb{R} of real numbers is isomorphic to its own dual.\nEvery locally compact group G has a good supply of irreducible unitary representations; for example, enough representations to distinguish the points of G (the [Gelfand–Raikov theorem](https://bliptext.com/articles/gelfand-raikov-theorem)).\nBy 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.\nFor example, there are many abelian [Banach–Lie groups](https://bliptext.com/articles/banach-manifold) for which every representation on Hilbert space is trivial.\n\n# Homotopy theory of topological groups\nTopological groups are special among all topological spaces, even in terms of their [homotopy type](https://bliptext.com/articles/homotopy-type).\nOne basic point is that a topological group G determines a path-connected topological space, the [classifying space](https://bliptext.com/articles/classifying-space)\nBG\n(which classifies [principal G-bundles](https://bliptext.com/articles/principal-bundle) over topological spaces, under mild hypotheses).\nThe group G is isomorphic in the [homotopy category](https://bliptext.com/articles/model-category) to the [loop space](https://bliptext.com/articles/loop-space) of\nBG\nSome of these restrictions hold in the broader context of [H-spaces](https://bliptext.com/articles/h-space).\nFor example, the [fundamental group](https://bliptext.com/articles/fundamental-group) of a topological group G is abelian.\n(More generally, the [Whitehead product](https://bliptext.com/articles/whitehead-product) on the homotopy groups of G is zero.)\nAlso, for any field k, the [cohomology](https://bliptext.com/articles/cohomology) ring\nH*(G,k)\nhas the structure of a [Hopf algebra](https://bliptext.com/articles/hopf-algebra).\nIn view of structure theorems on Hopf algebras by [Heinz Hopf](https://bliptext.com/articles/heinz-hopf) and [Armand Borel](https://bliptext.com/articles/armand-borel), this puts strong restrictions on the possible cohomology rings of topological groups.\nIn particular, if G is a path-connected topological group whose rational cohomology ring\nH*(G,\\mathbb{Q})\nis finite-dimensional in each degree, then this ring must be a free [graded-commutative](https://bliptext.com/articles/graded-commutative) algebra over \\mathbb{Q}, that is, the [tensor product](https://bliptext.com/articles/tensor-product-of-algebras) of a [polynomial ring](https://bliptext.com/articles/polynomial-ring) on generators of even degree with an [exterior algebra](https://bliptext.com/articles/exterior-algebra) on generators of odd degree.\nIn particular, for a connected Lie group G, the rational cohomology ring of G is an exterior algebra on generators of odd degree.\nMoreover, a connected Lie group G has a [maximal compact subgroup](https://bliptext.com/articles/maximal-compact-subgroup) K, which is unique up to conjugation, and the inclusion of K into G is a [homotopy equivalence](https://bliptext.com/articles/homotopy-equivalence).\nSo describing the homotopy types of Lie groups reduces to the case of compact Lie groups.\nFor example, the maximal compact subgroup of\nSL(2,\\mathbb{R})\nis the circle group\nSO(2)\n, and the homogeneous space\nSL(2,\\mathbb{R})/SO(2)\ncan be identified with the [hyperbolic plane](https://bliptext.com/articles/hyperbolic-plane).\nSince the hyperbolic plane is [contractible](https://bliptext.com/articles/contractible-space), the inclusion of the circle group into\nSL(2,\\mathbb{R})\nis a homotopy equivalence.\nFinally, compact connected Lie groups have been classified by [Wilhelm Killing](https://bliptext.com/articles/wilhelm-killing), [Élie Cartan](https://bliptext.com/articles/lie-cartan), and [Hermann Weyl](https://bliptext.com/articles/hermann-weyl).\nAs a result, there is an essentially complete description of the possible homotopy types of Lie groups.\nFor example, a compact connected Lie group of dimension at most 3 is either a torus, the group [SU(2)](https://bliptext.com/articles/su-2) ([diffeomorphic](https://bliptext.com/articles/diffeomorphic) to the 3-sphere\nS3\n), or its quotient group\nSU(2)/{±1} ≅ SO(3)\n(diffeomorphic to\nRP3\n).\n\n# Complete topological group\nInformation about convergence of nets and filters, such as definitions and properties, can be found in the article about [filters in topology](https://bliptext.com/articles/filters-in-topology).\n\n## Canonical uniformity on a commutative topological group\nThis article will henceforth assume that any topological group that we consider is an additive commutative topological group with identity element 0.\nThe **diagonal** of X is the set\nand for any containing 0, the **canonical entourage** or **canonical vicinities around N** is the set\nFor a topological group (X, \\tau), the **canonical uniformity** on X is the [uniform structure](https://bliptext.com/articles/uniform-space) induced by the set of all canonical entourages \\Delta(N) as N ranges over all neighborhoods of 0 in X.\nThat is, it is the upward closure of the following prefilter on X \\times X,\nwhere this prefilter forms what is known as a [base of entourages](https://bliptext.com/articles/base-of-entourages) of the canonical uniformity.\nFor a commutative additive group X, a fundamental system of entourages \\mathcal{B} is called a ****translation-invariant** uniformity** if for every if and only if for all A uniformity \\mathcal{B} is called **translation-invariant** if it has a base of entourages that is **translation-invariant**.\n \n The canonical uniformity on any commutative topological group is translation-invariant. \n The same canonical uniformity would result by using a neighborhood basis of the origin rather the filter of all neighborhoods of the origin. \n Every entourage \\Delta_X(N) contains the diagonal because 0 \\in N. \n If N is [symmetric](https://bliptext.com/articles/symmetric-set) (that is, -N = N) then \\Delta_X(N) is [symmetric](https://bliptext.com/articles/symmetric-set) (meaning that ) and\n The topology induced on X by the canonical uniformity is the same as the topology that X started with (that is, it is \\tau). \n \n\n## Cauchy prefilters and nets\nThe general theory of [uniform spaces](https://bliptext.com/articles/uniform-space) has its own definition of a \"Cauchy prefilter\" and \"Cauchy net.\"\nFor the canonical uniformity on X, these reduces down to the definition described below.\nSuppose is a net in X and is a net in Y. Make I \\times J into a directed set by declaring if and only if Then denotes the **product net**.\nIf X = Y then the image of this net under the addition map denotes the **sum** of these two nets:\nand similarly their **difference** is defined to be the image of the product net under the subtraction map:\nA [net](https://bliptext.com/articles/net-mathematics) in an additive topological group X is called a **Cauchy [net](https://bliptext.com/articles/net-mathematics)** if\nor equivalently, if for every neighborhood N of 0 in X, there exists some i_0 \\in I such that\nfor all indices\nA **[Cauchy sequence](https://bliptext.com/articles/cauchy-sequence)** is a Cauchy net that is a sequence.\nIf B is a subset of an additive group X and N is a set containing 0, thenB is said to be an **N-small set** or **small of order N** if\nA prefilter \\mathcal{B} on an additive topological group X called a **Cauchy prefilter** if it satisfies any of the following equivalent conditions:\n \n in X, where is a prefilter. \n in X, where is a prefilter equivalent to \n For every neighborhood N of 0 in X, \\mathcal{B} contains some N-small set (that is, there exists some such that ). \n \nand if X is commutative then also:\n For every neighborhood N of 0 in X, there exists some and some x \\in X such that \n \nSuppose \\mathcal{B} is a prefilter on a commutative topological group X and x \\in X. Then in X if and only if and \\mathcal{B} is Cauchy.\n\n## Complete commutative topological group\nRecall that for any a prefilter \\mathcal{C} **on S** is necessarily a subset of \\wp(S); that is,\nA subset S of a topological group X is called a **complete subset** if it satisfies any of the following equivalent conditions:\n \n Every Cauchy prefilter on S [converges](https://bliptext.com/articles/filters-in-topology) to at least one point of S. Every Cauchy net in S [converges](https://bliptext.com/articles/filters-in-topology) to at least one point of S; \n Every Cauchy filter \\mathcal{C} on S converges to at least one point of S. \n S is a [complete](https://bliptext.com/articles/complete-uniform-space) uniform space (under the point-set topology definition of \"[complete](https://bliptext.com/articles/complete-uniform-space) uniform space\") when S is endowed with the uniformity induced on it by the canonical uniformity of X; \n \nA 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.\nA commutative topological group X is called a **complete group** if any of the following equivalent conditions hold:\n \n X is complete as a subset of itself. \n Every Cauchy net in X [converges](https://bliptext.com/articles/net-mathematics) to at least one point of X. \n There exists a neighborhood of 0 in X that is also a complete subset of X. When endowed with its canonical uniformity, X becomes is a [complete uniform space](https://bliptext.com/articles/complete-uniform-space).\n \nA topological group is called **sequentially complete** if it is a **sequentially complete** subset of itself.\n**Neighborhood basis**: Suppose C is a completion of a commutative topological group X with and that \\mathcal{N} is a [neighborhood base](https://bliptext.com/articles/neighborhood-base) of the origin in X. Then the family of sets\nis a neighborhood basis at the origin in C.\n****\nLet X and Y be topological groups, and f : D \\to Y be a map.\nThen 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\n\n# Generalizations\nVarious generalizations of topological groups can be obtained by weakening the continuity conditions:\nc ∈ G\nthe two functions\nG → G\ndefined by\nx ↦ xc\nand\nx ↦ cx\nare continuous.\n\n# Citations","revisionCount":0},"hasMore":false,"nextCursor":null}

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