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Topological group

From Wikipedia, the free encyclopedia

Group that is a topological space with continuous group action

Algebraic structure →Group theory
Group theory

Basic notions

Group homomorphisms
Subgroup Normal subgroup Group action
Quotient group (Semi-)direct product Direct sum Free product Wreath product
kernel image
simple finite infinite continuous multiplicative additive cyclic abelian dihedral nilpotent solvable
Glossary of group theory List of group theory topics

Dihedral group D_n
Quaternion group Q

Frobenius group

Schur multiplier

Classification of finite simple groups

cyclic
alternating
Lie type
sporadic

Integers ( $\mathbb {Z}$ )
Free group

PSL(2, $\mathbb {Z}$ )
SL(2, $\mathbb {Z}$ )

Topological andLie groups

General linear GL(n)

Special linear SL(n)

Orthogonal O(n)

Euclidean E(n)

Special orthogonal SO(n)

Unitary U(n)

Special unitary SU(n)

Symplectic Sp(n)

Infinite dimensional Lie group

O(∞)
SU(∞)
Sp(∞)

Algebraic groups

Linear algebraic group

Reductive group

Abelian variety

Elliptic curve

v
t
e

Thereal numbers form a topological group underaddition

Inmathematics,topological groups are the combination ofgroups andtopological spaces, i.e. they are groups and topological spaces at the same time, such that thecontinuity condition for the group operations connects these two structures together and consequently they are not independent from each other.^[1]

Topological groups were studied extensively in the period of 1925 to 1940.Haar andWeil (respectively in 1933 and 1940) showed that theintegrals andFourier series are special cases of a construct that can be defined on a very wide class of topological groups.^[2]

Topological groups, along withcontinuous group actions, are used to study continuoussymmetries, which have many applications, for example,in physics. Infunctional analysis, everytopological vector space is an additive topological group with the additional property that scalar multiplication is continuous; consequently, many results from the theory of topological groups can be applied to functional analysis.

Formal definition

Atopological group,G, is atopological space that is also a group such that the group operation (in this case product):

⋅ :G ×G →G, (x,y) ↦xy

and the inversion map:

⁻¹ :G →G,x ↦x⁻¹

arecontinuous.^{[note 1]} HereG ×G is viewed as a topological space with theproduct topology. Such a topology is said to becompatible with the group operations and is called agroup topology.

Checking continuity

The product map is continuous if and only if for anyx,y ∈G and any neighborhoodW ofxy inG, there exist neighborhoodsU ofx andV ofy inG such thatU ⋅V ⊆W, whereU ⋅V := {u ⋅v :u ∈U,v ∈V}. The inversion map is continuous if and only if for anyx ∈G and any neighborhoodV ofx⁻¹ inG, there exists a neighborhoodU ofx inG such thatU⁻¹ ⊆V, whereU⁻¹ := {u⁻¹ :u ∈U}.

To show that a topology is compatible with the group operations, it suffices to check that the map

G ×G →G,(x,y) ↦xy⁻¹

is continuous. Explicitly, this means that for anyx,y ∈G and any neighborhoodW inG ofxy⁻¹, there exist neighborhoodsU ofx andV ofy inG such thatU ⋅ (V⁻¹) ⊆W.

Additive notation

This definition used notation for multiplicative groups; the equivalent for additive groups would be that the following two operations are continuous:

+ :G ×G →G, (x,y) ↦x +y

− :G →G,x ↦ −x.

Hausdorffness

Although not part of this definition, many authors^[3] require that the topology onG beHausdorff. 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 ofcategory theory, topological groups can be defined concisely asgroup objects in thecategory of topological spaces, in the same way that ordinary groups are group objects in thecategory 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

Ahomomorphism of topological groups means a continuousgroup homomorphismG →H. Topological groups, together with their homomorphisms, form acategory. A group homomorphism between topological groups is continuous if and only if it is continuous atsome point.^[4]

Anisomorphism of topological groups is agroup isomorphism that is also ahomeomorphism 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 thediscrete topology; such groups are calleddiscrete groups. In this sense, the theory of topological groups subsumes that of ordinary groups. Theindiscrete topology (i.e. the trivial topology) also makes every group into a topological group.

Thereal numbers, $\mathbb {R}$ with the usual topology form a topological group under addition.Euclideann-space $\mathbb {R}$ ⁿ is also a topological group under addition, and more generally, everytopological vector space forms an (abelian) topological group. Some other examples ofabelian topological groups are thecircle groupS¹, or thetorus(S¹)ⁿ for any natural numbern.

Theclassical groups are important examples of non-abelian topological groups. For instance, thegeneral linear groupGL(n, $\mathbb {R}$ ) of all invertiblen-by-nmatrices with real entries can be viewed as a topological group with the topology defined by viewingGL(n, $\mathbb {R}$ ) as asubspace of Euclidean space $\mathbb {R}$ ^n×n. Another classical group is theorthogonal groupO(n), the group of alllinear maps from $\mathbb {R}$ ⁿ to itself that preserve thelength of all vectors. The orthogonal group iscompact as a topological space. Much ofEuclidean geometry can be viewed as studying the structure of the orthogonal group, or the closely related groupO(n) ⋉ $\mathbb {R}$ ⁿ ofisometries of $\mathbb {R}$ ⁿ.

The groups mentioned so far are allLie groups, meaning that they aresmooth manifolds in such a way that the group operations aresmooth, not just continuous. Lie groups are the best-understood topological groups; many questions about Lie groups can be converted to purely algebraic questions aboutLie algebras and then solved.

An example of a topological group that is not a Lie group is the additive group $\mathbb {Q}$ ofrational numbers, with the topology inherited from $\mathbb {R}$ . This is acountable space, and it does not have the discrete topology. An important example fornumber theory is the group $\mathbb {Z}$ _p ofp-adic integers, for aprime numberp, meaning theinverse limit of the finite groups $\mathbb {Z}$ /pⁿ asn goes to infinity. The group $\mathbb {Z}$ _p is well behaved in that it is compact (in fact, homeomorphic to theCantor set), but it differs from (real) Lie groups in that it istotally disconnected. More generally, there is a theory ofp-adic Lie groups, including compact groups such asGL(n, $\mathbb {Z}$ _p) as well aslocally compact groups such asGL(n, $\mathbb {Q}$ _p), where $\mathbb {Q}$ _p is the locally compactfield ofp-adic numbers.

The group $\mathbb {Z}$ _p is apro-finite group; it is isomorphic to a subgroup of the product $\prod _{n\geq 1}\mathbb {Z} /p^{n}$ in such a way that its topology is induced by the product topology, where the finite groups $\mathbb {Z} /p^{n}$ are given the discrete topology. Another large class of pro-finite groups important in number theory areabsolute Galois groups.

Some topological groups can be viewed asinfinite dimensional Lie groups; this phrase is best understood informally, to include several different families of examples. For example, atopological vector space, such as aBanach space orHilbert space, is an abelian topological group under addition. Some other infinite-dimensional groups that have been studied, with varying degrees of success, areloop groups,Kac–Moody groups,Diffeomorphism groups,homeomorphism groups, andgauge groups.

In everyBanach algebra with multiplicative identity, the set of invertible elements forms a topological group under multiplication. For example, the group of invertiblebounded operators on a Hilbert space arises this way.

Properties

Translation invariance

Every topological group's topology istranslation invariant, 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 $S {\displaystyle S}$ isopen (resp.closed) in $G {\displaystyle G}$ if and only if this is true of its left translation $aS:=\{as:s\in S\}$ and right translation $Sa:=\{sa:s\in S\}.$ If ${\mathcal {N}}$ is aneighborhood basis of the identity element in a topological group $G {\displaystyle G}$ then for all $x\in X,$ $x{\mathcal {N}}:=\{xN:N\in {\mathcal {N}}\}$ is a neighborhood basis of $x {\displaystyle x}$ in $G . {\displaystyle G.}$ ^[4] In particular, any group topology on a topological group is completely determined by any neighborhood basis at the identity element. If $S {\displaystyle S}$ is any subset of $G {\displaystyle G}$ and $U {\displaystyle U}$ is an open subset of $G, {\displaystyle G,}$ then $SU:=\{su:s\in S,u\in U\}$ is an open subset of $G . {\displaystyle G.}$ ^[4]

Symmetric neighborhoods

The inversion operation $g\mapsto g^{-1}$ on a topological group $G {\displaystyle G}$ is a homeomorphism from $G {\displaystyle G}$ to itself.

A subset $S\subseteq G$ is said to besymmetric if $S^{-1}=S,$ where $S^{-1}:=\left\{s^{-1}:s\in S\right\}.$ The closure of every symmetric set in a commutative topological group is symmetric.^[4] IfS is any subset of a commutative topological groupG, then the following sets are also symmetric:S⁻¹ ∩S,S⁻¹ ∪S, andS⁻¹S.^[4]

For any neighborhoodN in a commutative topological groupG of the identity element, there exists a symmetric neighborhoodM of the identity element such thatM⁻¹M ⊆N, where note thatM⁻¹M is necessarily a symmetric neighborhood of the identity element.^[4] Thus every topological group has a neighborhood basis at the identity element consisting of symmetric sets.

IfG is alocally compact commutative group, then for any neighborhoodN inG of the identity element, there exists a symmetric relatively compact neighborhoodM of the identity element such thatclM ⊆N (whereclM is symmetric as well).^[4]

Uniform space

Every topological group can be viewed as auniform space in two ways; theleft uniformity turns all left multiplications into uniformly continuous maps while theright uniformity turns all right multiplications into uniformly continuous maps.^[5]IfG is not abelian, then these two need not coincide. The uniform structures allow one to talk about notions such ascompleteness,uniform continuity anduniform convergence on topological groups.

Separation properties

IfU is an open subset of a commutative topological groupG andU contains a compact setK, then there exists a neighborhoodN of the identity element such thatKN ⊆U.^[4]

As a uniform space, every commutative topological group iscompletely regular. Consequently, for a multiplicative topological groupG with identity element 1, the following are equivalent:^[4]

G is a T₀-space (Kolmogorov);
G is a T₂-space (Hausdorff);
G is a T_31⁄2 (Tychonoff);
{ 1 } is closed inG;
{ 1 } :=∩N ∈ 𝒩N, where𝒩 is a neighborhood basis of the identity element inG;
for any $x\in G$ such that $x\neq 1,$ there exists a neighborhoodU inG 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 anisolated point.^[4]

IfG is not Hausdorff, then one can obtain a Hausdorff group by passing to the quotient groupG/K, whereK is theclosure of the identity.^[6] This is equivalent to taking theKolmogorov quotient ofG.

Metrisability

LetG be a topological group. As with any topological space, we say thatG ismetrisable if and only if there exists a metricd onG, which induces the same topology on $G {\displaystyle G}$ . A metricd onG is called

left-invariant (resp.right-invariant) if and only if $d(ax_{1},ax_{2})=d(x_{1},x_{2})$ (resp. $d(x_{1}a,x_{2}a)=d(x_{1},x_{2})$ ) for all $a,x_{1},x_{2}\in G$ (equivalently, $d {\displaystyle d}$ is left-invariant just in case the map $x\mapsto ax$ is anisometry from $(G,d)$ to itself for each $a\in G$ ).
proper if and only if all open balls, $B(r)=\{g\in G\mid d(g,1)<r\}$ for $r>0$ , are pre-compact.

TheBirkhoff–Kakutani theorem (named after mathematiciansGarrett Birkhoff andShizuo Kakutani) states that the following three conditions on a topological groupG are equivalent:^[7]

G is (Hausdorff and)first countable (equivalently: the identity element 1 is closed inG, and there is a countablebasis of neighborhoods for 1 inG).
G ismetrisable (as a topological space).
There is a left-invariant metric onG that induces the given topology onG.
There is a right-invariant metric onG that induces the given topology onG.

Furthermore, the following are equivalent for any topological groupG:

G is asecond countable locally compact (Hausdorff) space.
G is aPolish,locally compact (Hausdorff) space.
G is properlymetrisable (as a topological space).
There is a left-invariant, proper metric onG that induces the given topology onG.

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.^[8] An alternative approach was made byUffe Haagerup and Agata Przybyszewska in 2006,^[9]the idea of the which is as follows:One relies on the construction of a left-invariant metric, $d_{0}$ , as in the case offirst 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,U, of radius 1 under multiplication yields aclopen subgroup,H, ofG, on which the metric $d_{0}$ is proper. SinceH is open andG issecond countable, the subgroup has at most countably many cosets. One now uses this sequence of cosets and the metric onH to construct a proper metric onG.

Subgroups

Everysubgroup of a topological group is itself a topological group when given thesubspace topology. Every open subgroupH is also closed inG, since the complement ofH is the open set given by the union of open setsgH forg ∈G \H. IfH is a subgroup ofG then the closure ofH is also a subgroup. Likewise, ifH is a normal subgroup ofG, the closure ofH is normal inG.

Quotients and normal subgroups

IfH is a subgroup ofG, the set of leftcosetsG/H with thequotient topology is called ahomogeneous space forG. The quotient map $q:G\to G/H$ is alwaysopen. For example, for a positive integern, thesphereSⁿ is a homogeneous space for therotation groupSO(n+1) in $\mathbb {R}$ ⁿ⁺¹, withSⁿ = SO(n+1)/SO(n). A homogeneous spaceG/H is Hausdorff if and only ifH is closed inG.^[10] Partly for this reason, it is natural to concentrate on closed subgroups when studying topological groups.

IfH is anormal subgroup ofG, then thequotient groupG/H becomes a topological group when given the quotient topology. It is Hausdorff if and only ifH is closed inG. For example, the quotient group $\mathbb {R} /\mathbb {Z}$ is isomorphic to the circle groupS¹.

In any topological group, theidentity component (i.e., theconnected component containing the identity element) is a closed normal subgroup. IfC is the identity component anda is any point ofG, then the left cosetaC is the component ofG containinga. So the collection of all left cosets (or right cosets) ofC inG is equal to the collection of all components ofG. It follows that the quotient groupG/C istotally disconnected.^[11]

Closure and compactness

In any commutative topological group, the product (assuming the group is multiplicative)KC of a compact setK and a closed setC is a closed set.^[4] Furthermore, for any subsetsR andS ofG,(clR)(clS) ⊆ cl (RS).^[4]

IfH is a subgroup of a commutative topological groupG and ifN is a neighborhood inG of the identity element such thatH ∩ clN is closed, thenH is closed.^[4] Every discrete subgroup of a Hausdorff commutative topological group is closed.^[4]

Isomorphism theorems

Theisomorphism 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 ${\tilde {f}}:G/\ker f\to \mathrm {Im} (f)$ 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 thecategory of topological groups. For example, consider the identity map from the set of real numbers equipped with the discrete topology to the set of real numbers equipped with the Euclidean topology. This is a group homomorphism, and it is continuous because any function out of a discrete space is continuous, but it is not an isomorphism of topological groups because its inverse is not continuous.

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 fromG/ker(f) toim(f) is an isomorphism if and only if the mapf is open onto its image.^[12]

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 smoothsubmanifold.

Hilbert's fifth problem asked whether a topological groupG that is atopological manifold must be a Lie group. In other words, doesG have the structure of a smooth manifold, making the group operations smooth? As shown byAndrew Gleason,Deane Montgomery, andLeo Zippin, the answer to this problem is yes.^[13] In fact,G has areal analytic structure. Using the smooth structure, one can define the Lie algebra ofG, an object oflinear algebra that determines aconnected groupG up tocovering 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, everycompact group (understood to be Hausdorff) is an inverse limit of compact Lie groups. (One important case is an inverse limit of finite groups, called aprofinite group. For example, the group $\mathbb {Z}$ _p ofp-adic integers and theabsolute Galois group of a field are profinite groups.) Furthermore, every connected locally compact group is an inverse limit of connected Lie groups.^[14] At the other extreme, a totally disconnected locally compact group always contains a compact open subgroup, which is necessarily a profinite group.^[15] (For example, the locally compact groupGL(n, $\mathbb {Q}$ _p) contains the compact open subgroupGL(n, $\mathbb {Z}$ _p), which is the inverse limit of the finite groupsGL(n, $\mathbb {Z}$ /p^r) asr' goes to infinity.)

Representations of compact or locally compact groups

Anaction of a topological groupG on a topological spaceX is agroup action ofG onX such that the corresponding functionG ×X →X is continuous. Likewise, arepresentation of a topological groupG on a real or complex topological vector spaceV is a continuous action ofG onV such that for eachg ∈G, the mapv ↦gv fromV to itself is linear.

Group actions and representation theory are particularly well understood for compact groups, generalizing what happens forfinite groups. For example, every finite-dimensional (real or complex) representation of a compact group is adirect sum ofirreducible representations. An infinite-dimensionalunitary 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 thePeter–Weyl theorem.^[16] For example, the theory ofFourier series describes the decomposition of the unitary representation of the circle groupS¹ on the complex Hilbert spaceL²(S¹). The irreducible representations ofS¹ are all 1-dimensional, of the formz ↦zⁿ for integersn (whereS¹ is viewed as a subgroup of the multiplicative group $\mathbb {C}$ *). Each of these representations occurs with multiplicity 1 inL²(S¹).

The irreducible representations of all compact connected Lie groups have been classified. In particular, thecharacter of each irreducible representation is given by theWeyl character formula.

More generally, locally compact groups have a rich theory ofharmonic analysis, because they admit a natural notion ofmeasure andintegral, given by theHaar measure. Every unitary representation of a locally compact group can be described as adirect integral of irreducible unitary representations. (The decomposition is essentially unique ifG is ofType I, which includes the most important examples such as abelian groups andsemisimple Lie groups.^[17]) A basic example is theFourier transform, which decomposes the action of the additive group $\mathbb {R}$ on the Hilbert spaceL²( $\mathbb {R}$ ) as a direct integral of the irreducible unitary representations of $\mathbb {R}$ . The irreducible unitary representations of $\mathbb {R}$ are all 1-dimensional, of the formx ↦e^2πiax fora ∈ $\mathbb {R}$ .

The irreducible unitary representations of a locally compact group may be infinite-dimensional. A major goal of representation theory, related to theLanglands classification ofadmissible representations, is to find theunitary dual (the space of all irreducible unitary representations) for the semisimple Lie groups. The unitary dual is known in many cases such asSL(2, $\mathbb {R}$ ), but not all.

For alocally compact abelian groupG, every irreducible unitary representation has dimension 1. In this case, the unitary dual ${\hat {G}}$ is a group, in fact another locally compact abelian group.Pontryagin duality states that for a locally compact abelian groupG, the dual of ${\hat {G}}$ is the original groupG. For example, the dual group of the integers $\mathbb {Z}$ is the circle groupS¹, while the group $\mathbb {R}$ of real numbers is isomorphic to its own dual.

Every locally compact groupG has a good supply of irreducible unitary representations; for example, enough representations to distinguish the points ofG (theGelfand–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 abelianBanach–Lie groups for which every representation on Hilbert space is trivial.^[18]

Homotopy theory of topological groups

Topological groups are special among all topological spaces, even in terms of theirhomotopy type. One basic point is that a topological groupG determines a path-connected topological space, theclassifying spaceBG (which classifiesprincipalG-bundles over topological spaces, under mild hypotheses). The groupG is isomorphic in thehomotopy category to theloop space ofBG; that implies various restrictions on the homotopy type ofG.^[19] Some of these restrictions hold in the broader context ofH-spaces.

For example, thefundamental group of a topological groupG is abelian. (More generally, theWhitehead product on the homotopy groups ofG is zero.) Also, for any fieldk, thecohomology ringH*(G,k) has the structure of aHopf algebra. In view of structure theorems on Hopf algebras byHeinz Hopf andArmand Borel, this puts strong restrictions on the possible cohomology rings of topological groups. In particular, ifG is a path-connected topological group whose rational cohomology ringH*(G, $\mathbb {Q}$ ) is finite-dimensional in each degree, then this ring must be a freegraded-commutative algebra over $\mathbb {Q}$ , that is, thetensor product of apolynomial ring on generators of even degree with anexterior algebra on generators of odd degree.^[20]

In particular, for a connected Lie groupG, the rational cohomology ring ofG is an exterior algebra on generators of odd degree. Moreover, a connected Lie groupG has amaximal compact subgroupK, which is unique up to conjugation, and the inclusion ofK intoG is ahomotopy equivalence. So describing the homotopy types of Lie groups reduces to the case of compact Lie groups. For example, the maximal compact subgroup ofSL(2, $\mathbb {R}$ ) is the circle groupSO(2), and the homogeneous spaceSL(2, $\mathbb {R}$ )/SO(2) can be identified with thehyperbolic plane. Since the hyperbolic plane iscontractible, the inclusion of the circle group intoSL(2, $\mathbb {R}$ ) is a homotopy equivalence.

Finally, compact connected Lie groups have been classified byWilhelm Killing,Élie Cartan, andHermann Weyl. As a result, there is an essentially complete description of the possible homotopy types of Lie groups. For example, a compact connected Lie group of dimension at most 3 is either a torus, the groupSU(2) (diffeomorphic to the 3-sphereS³), or its quotient groupSU(2)/{±1} ≅SO(3) (diffeomorphic toRP³).

Complete topological group

See also:Complete uniform space

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

Canonical uniformity on a commutative topological group

Main article:Uniform space

This article will henceforth assume that any topological group that we consider is an additive commutative topological group with identity element $0. {\displaystyle 0.}$

Thediagonal of $X {\displaystyle X}$ is the set $\Delta _{X}:=\{(x,x):x\in X\}$ and for any $N\subseteq X$ containing $0,$ thecanonical entourage orcanonical vicinities around $N {\displaystyle N}$ is the set $\Delta _{X}(N):=\{(x,y)\in X\times X:x-y\in N\}=\bigcup _{y\in X}[(y+N)\times \{y\}]=\Delta _{X}+(N\times \{0\})$

For a topological group $(X,\tau ),$ thecanonical uniformity^[21] on $X {\displaystyle X}$ is theuniform structure induced by the set of all canonical entourages $\Delta (N)$ as $N {\displaystyle N}$ ranges over all neighborhoods of $0 {\displaystyle 0}$ in $X . {\displaystyle X.}$

That is, it is the upward closure of the following prefilter on $X\times X,$ $\left\{\Delta (N):N{\text{ is a neighborhood of }}0{\text{ in }}X\right\}$ where this prefilter forms what is known as abase of entourages of the canonical uniformity.

For a commutative additive group $X, {\displaystyle X,}$ a fundamental system of entourages ${\mathcal {B}}$ is called atranslation-invariant uniformity if for every $B\in {\mathcal {B}},$ $(x,y)\in B$ if and only if $(x+z,y+z)\in B$ for all $x,y,z\in X.$ A uniformity ${\mathcal {B}}$ is calledtranslation-invariant if it has a base of entourages that is translation-invariant.^[22]

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 $\Delta _{X}:=\Delta _{X}(\{0\})=\{(x,x):x\in X\}$ because $0\in N.$
If $N {\displaystyle N}$ issymmetric (that is, $-N=N$ ) then $\Delta _{X}(N)$ is symmetric (meaning that $\Delta _{X}(N)^{\operatorname {op} }=\Delta _{X}(N)$ ) and $\Delta _{X}(N)\circ \Delta _{X}(N)=\{(x,z):{\text{ there exists }}y\in X{\text{ such that }}x,z\in y+N\}=\bigcup _{y\in X}[(y+N)\times (y+N)]=\Delta _{X}+(N\times N).$
The topology induced on $X {\displaystyle X}$ by the canonical uniformity is the same as the topology that $X {\displaystyle X}$ started with (that is, it is $\tau$ ).

Cauchy prefilters and nets

Main articles:Filters in topology andNet (mathematics)

The general theory ofuniform spaces has its own definition of a "Cauchy prefilter" and "Cauchy net." For the canonical uniformity on $X, {\displaystyle X,}$ these reduces down to the definition described below.

Suppose $x_{\bullet }=\left(x_{i}\right)_{i\in I}$ is a net in $X {\displaystyle X}$ and $y_{\bullet }=\left(y_{j}\right)_{j\in J}$ is a net in $Y . {\displaystyle Y.}$ Make $I\times J$ into a directed set by declaring $(i,j)\leq \left(i_{2},j_{2}\right)$ if and only if $i\leq i_{2}{\text{ and }}j\leq j_{2}.$ Then^[23] $x_{\bullet }\times y_{\bullet }:=\left(x_{i},y_{j}\right)_{(i,j)\in I\times J}$ denotes theproduct net. If $X=Y$ then the image of this net under the addition map $X\times X\to X$ denotes thesum of these two nets: $x_{\bullet }+y_{\bullet }:=\left(x_{i}+y_{j}\right)_{(i,j)\in I\times J}$ and similarly theirdifference is defined to be the image of the product net under the subtraction map: $x_{\bullet }-y_{\bullet }:=\left(x_{i}-y_{j}\right)_{(i,j)\in I\times J}.$

Anet $x_{\bullet }=\left(x_{i}\right)_{i\in I}$ in an additive topological group $X {\displaystyle X}$ is called aCauchy net if^[24] $\left(x_{i}-x_{j}\right)_{(i,j)\in I\times I}\to 0{\text{ in }}X$ or equivalently, if for every neighborhood $N {\displaystyle N}$ of $0 {\displaystyle 0}$ in $X, {\displaystyle X,}$ there exists some $i_{0}\in I$ such that $x_{i}-x_{j}\in N$ for all indices $i,j\geq i_{0}.$

ACauchy sequence is a Cauchy net that is a sequence.

If $B {\displaystyle B}$ is a subset of an additive group $X {\displaystyle X}$ and $N {\displaystyle N}$ is a set containing $0,$ then $B {\displaystyle B}$ is said to be an $N {\displaystyle N}$ -small set orsmall of order $N {\displaystyle N}$ if $B-B\subseteq N.$ ^[25]

A prefilter ${\mathcal {B}}$ on an additive topological group $X {\displaystyle X}$ called aCauchy prefilter if it satisfies any of the following equivalent conditions:

${\mathcal {B}}-{\mathcal {B}}\to 0$ in $X, {\displaystyle X,}$ where ${\mathcal {B}}-{\mathcal {B}}:=\{B-C:B,C\in {\mathcal {B}}\}$ is a prefilter.
$\{B-B:B\in {\mathcal {B}}\}\to 0$ in $X, {\displaystyle X,}$ where $\{B-B:B\in {\mathcal {B}}\}$ is a prefilter equivalent to ${\mathcal {B}}-{\mathcal {B}}.$
For every neighborhood $N {\displaystyle N}$ of $0 {\displaystyle 0}$ in $X, {\displaystyle X,}$ ${\mathcal {B}}$ contains some $N {\displaystyle N}$ -small set (that is, there exists some $B\in {\mathcal {B}}$ such that $B-B\subseteq N$ ).^[25]

and if $X {\displaystyle X}$ is commutative then also:

For every neighborhood $N {\displaystyle N}$ of $0 {\displaystyle 0}$ in $X, {\displaystyle X,}$ there exists some $B\in {\mathcal {B}}$ and some $x\in X$ such that $B\subseteq x+N.$ ^[25]

It suffices to check any of the above condition for any givenneighborhood basis of $0 {\displaystyle 0}$ in $X . {\displaystyle X.}$

Suppose ${\mathcal {B}}$ is a prefilter on a commutative topological group $X {\displaystyle X}$ and $x\in X.$ Then ${\mathcal {B}}\to x$ in $X {\displaystyle X}$ if and only if $x\in \operatorname {cl} {\mathcal {B}}$ and ${\mathcal {B}}$ is Cauchy.^[23]

Complete commutative topological group

Main article:Complete uniform space

Recall that for any $S\subseteq X,$ a prefilter ${\mathcal {C}}$ on $S {\displaystyle S}$ is necessarily a subset of $\wp (S)$ ; that is, ${\mathcal {C}}\subseteq \wp (S).$

A subset $S {\displaystyle S}$ of a topological group $X {\displaystyle X}$ is called acomplete subset if it satisfies any of the following equivalent conditions:

Every Cauchy prefilter ${\mathcal {C}}\subseteq \wp (S)$ on $S {\displaystyle S}$ converges to at least one point of $S . {\displaystyle S.}$
- If $X {\displaystyle X}$ is Hausdorff then every prefilter on $S {\displaystyle S}$ will converge to at most one point of $X . {\displaystyle X.}$ But if $X {\displaystyle X}$ is not Hausdorff then a prefilter may converge to multiple points in $X . {\displaystyle X.}$ The same is true for nets.
Every Cauchy net in $S {\displaystyle S}$ converges to at least one point of $S {\displaystyle S}$ ;
Every Cauchy filter ${\mathcal {C}}$ on $S {\displaystyle S}$ converges to at least one point of $S . {\displaystyle S.}$
$S {\displaystyle S}$ is acomplete uniform space (under the point-set topology definition of "complete uniform space") when $S {\displaystyle S}$ is endowed with the uniformity induced on it by the canonical uniformity of $X {\displaystyle X}$ ;

A subset $S {\displaystyle S}$ is called asequentially complete subset if every Cauchy sequence in $S {\displaystyle S}$ (or equivalently, every elementary Cauchy filter/prefilter on $S {\displaystyle S}$ ) converges to at least one point of $S . {\displaystyle S.}$

Importantly,convergence outside of $S {\displaystyle S}$ is allowed: If $X {\displaystyle X}$ is not Hausdorff and if every Cauchy prefilter on $S {\displaystyle S}$ converges to some point of $S, {\displaystyle S,}$ then $S {\displaystyle S}$ will be complete even if some or all Cauchy prefilters on $S {\displaystyle S}$ also converge to points(s) in the complement $X\setminus S.$ In short, there is no requirement that these Cauchy prefilters on $S {\displaystyle S}$ convergeonly to points in $S . {\displaystyle S.}$ The same can be said of the convergence of Cauchy nets in $S . {\displaystyle S.}$
- As a consequence, if a commutative topological group $X {\displaystyle X}$ isnotHausdorff, then every subset of the closure of $\{0\},$ say $S\subseteq \operatorname {cl} \{0\},$ is complete (since it is clearly compact and every compact set is necessarily complete). So in particular, if $S\neq \varnothing$ (for example, if $S {\displaystyle S}$ a is singleton set such as $S=\{0\}$ ) then $S {\displaystyle S}$ would be complete even thoughevery Cauchy net in $S {\displaystyle S}$ (and every Cauchy prefilter on $S {\displaystyle S}$ ), converges toevery point in $\operatorname {cl} \{0\}$ (include those points in $\operatorname {cl} \{0\}$ that are not in $S {\displaystyle S}$ ).
- This example also shows that complete subsets (indeed, even compact subsets) of a non-Hausdorff space may fail to be closed (for example, if $\varnothing \neq S\subseteq \operatorname {cl} \{0\}$ then $S {\displaystyle S}$ is closed if and only if $S=\operatorname {cl} \{0\}$ ).

A commutative topological group $X {\displaystyle X}$ is called acomplete group if any of the following equivalent conditions hold:

$X {\displaystyle X}$ is complete as a subset of itself.
Every Cauchy net in $X {\displaystyle X}$ converges to at least one point of $X . {\displaystyle X.}$
There exists a neighborhood of $0 {\displaystyle 0}$ in $X {\displaystyle X}$ that is also a complete subset of $X . {\displaystyle X.}$ ^[25]
- This implies that every locally compact commutative topological group is complete.
When endowed with its canonical uniformity, $X {\displaystyle X}$ becomes is acomplete uniform space.
- In the general theory ofuniform spaces, a uniform space is called acomplete uniform space if each Cauchyfilter in $X {\displaystyle X}$ converges in $(X,\tau )$ to some point of $X . {\displaystyle X.}$

A topological group is calledsequentially complete if it is a sequentially complete subset of itself.

Neighborhood basis: Suppose $C {\displaystyle C}$ is a completion of a commutative topological group $X {\displaystyle X}$ with $X\subseteq C$ and that ${\mathcal {N}}$ is aneighborhood base of the origin in $X . {\displaystyle X.}$ Then the family of sets $\left\{\operatorname {cl} _{C}N:N\in {\mathcal {N}}\right\}$ is a neighborhood basis at the origin in $C . {\displaystyle C.}$ ^[23]

Uniform continuity

Let $X {\displaystyle X}$ and $Y {\displaystyle Y}$ be topological groups, $D\subseteq X,$ and $f:D\to Y$ be a map. Then $f:D\to Y$ isuniformly continuous if for every neighborhood $U {\displaystyle U}$ of the origin in $X, {\displaystyle X,}$ there exists a neighborhood $V {\displaystyle V}$ of the origin in $Y {\displaystyle Y}$ such that for all $x,y\in D,$ if $y-x\in U$ then $f(y)-f(x)\in V.$

Generalizations

Various generalizations of topological groups can be obtained by weakening the continuity conditions:^[26]

Asemitopological group is a groupG with a topology such that for eachc ∈G the two functionsG →G defined byx ↦xc andx ↦cx are continuous.
Aquasitopological group is a semitopological group in which the function mapping elements to their inverses is also continuous.
Aparatopological group is a group with a topology such that the group operation is continuous.

See also

Algebraic group – Algebraic variety with a group structure
Complete field – algebraic structure that is complete relative to a metricPages displaying wikidata descriptions as a fallback
Compact group – Topological group with compact topology
Complete topological vector space – A TVS where points that get progressively closer to each other will always converge to a point
Lie group – Group that is also a differentiable manifold with group operations that are smooth
Locally compact field
Locally compact group – topological group for which the underlying topology is locally compact and Hausdorff, so that the Haar measure can be definedPages displaying wikidata descriptions as a fallback
Locally compact quantum group – relatively new C*-algebraic approach toward quantum groupsPages displaying wikidata descriptions as a fallback
Profinite group – Topological group that is in a certain sense assembled from a system of finite groups
Ordered topological vector space
Topological abelian group – topological group whose group is abelianPages displaying wikidata descriptions as a fallback
Topological field – Algebraic structure with addition, multiplication, and divisionPages displaying short descriptions of redirect targets
Topological module
Topological ring – ring where ring operations are continuousPages displaying wikidata descriptions as a fallback
Topological semigroup – semigroup with continuous operationPages displaying wikidata descriptions as a fallback
Topological vector space – Vector space with a notion of nearness

Notes

^i.e. Continuous means that for any open setU ⊆G,f⁻¹(U) is open in the domaindomf off.

Citations

^Pontrjagin 1946, p. 52.
^Hewitt & Ross 1979, p. 1.
^Armstrong 1997, p. 73;Bredon 1997, p. 51
^^a ^b ^c ^d ^e ^f ^g ^h ⁱ ^j ^k ^l ^m ⁿNarici & Beckenstein 2011, pp. 19–45.
^Bourbaki 1998, section III.3.
^Bourbaki 1998, section III.2.7.
^Montgomery & Zippin 1955, section 1.22.
^Struble, Raimond A. (1974)."Metrics in locally compact groups".Compositio Mathematica.28 (3):217–222.
^Haagerup, Uffe; Przybyszewska, Agata (2006),Proper metrics on locally compact groups, and proper affine isometric actions on,CiteSeerX 10.1.1.236.827
^Bourbaki 1998, section III.2.5.
^Bourbaki 1998, section I.11.5.
^Bourbaki 1998, section III.2.8.
^Montgomery & Zippin 1955, section 4.10.
^Montgomery & Zippin 1955, section 4.6.
^Bourbaki 1998, section III.4.6.
^Hewitt & Ross 1970, Theorem 27.40.
^Mackey 1976, section 2.4.
^Banaszczyk 1983.
^Hatcher 2001, Theorem 4.66.
^Hatcher 2001, Theorem 3C.4.
^Edwards 1995, p. 61.
^Schaefer & Wolff 1999, pp. 12–19.
^^a ^b ^cNarici & Beckenstein 2011, pp. 47–66.
^Narici & Beckenstein 2011, p. 48.
^^a ^b ^c ^dNarici & Beckenstein 2011, pp. 48–51.
^Arhangel'skii & Tkachenko 2008, p. 12.

References

Arhangel'skii, Alexander; Tkachenko, Mikhail (2008).Topological Groups and Related Structures.World Scientific.ISBN 978-90-78677-06-2.MR 2433295.
Armstrong, Mark A. (1997).Basic Topology (1st ed.).Springer-Verlag.ISBN 0-387-90839-0.MR 0705632.
Banaszczyk, Wojciech (1983),"On the existence of exotic Banach–Lie groups",Mathematische Annalen,264 (4):485–493,doi:10.1007/BF01456956,MR 0716262,S2CID 119755117
Bourbaki, Nicolas (1998),General Topology. Chapters 1–4,Springer-Verlag,ISBN 3-540-64241-2,MR 1726779
Folland, Gerald B. (1995),A Course in Abstract Harmonic Analysis,CRC Press,ISBN 0-8493-8490-7
Bredon, Glen E. (1997).Topology and Geometry. Graduate Texts in Mathematics (1st ed.).Springer-Verlag.ISBN 0-387-97926-3.MR 1700700.
Hatcher, Allen (2001),Algebraic Topology,Cambridge University Press,ISBN 0-521-79540-0,MR 1867354
Edwards, Robert E. (1995).Functional Analysis: Theory and Applications. New York: Dover Publications.ISBN 978-0-486-68143-6.OCLC 30593138.
Hewitt, Edwin;Ross, Kenneth A. (1979),Abstract Harmonic Analysis, vol. 1 (2nd ed.),Springer-Verlag,ISBN 978-0387941905,MR 0551496
Hewitt, Edwin;Ross, Kenneth A. (1970),Abstract Harmonic Analysis, vol. 2,Springer-Verlag,ISBN 978-0387048321,MR 0262773
Mackey, George W. (1976),The Theory of Unitary Group Representations,University of Chicago Press,ISBN 0-226-50051-9,MR 0396826
Montgomery, Deane;Zippin, Leo (1955),Topological Transformation Groups, New York, London:Interscience Publishers,MR 0073104
Narici, Lawrence; Beckenstein, Edward (2011).Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press.ISBN 978-1584888666.OCLC 144216834.
Pontrjagin, Leon (1946).Topological Groups.Princeton University Press.
Pontryagin, Lev S. (1986).Topological Groups. trans. from Russian by Arlen Brown and P.S.V. Naidu (3rd ed.). New York: Gordon and Breach Science Publishers.ISBN 2-88124-133-6.MR 0201557.
Porteous, Ian R. (1981).Topological Geometry (2nd ed.).Cambridge University Press.ISBN 0-521-23160-4.MR 0606198.
Schaefer, Helmut H.; Wolff, Manfred P. (1999).Topological Vector Spaces.GTM. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer.ISBN 978-1-4612-7155-0.OCLC 840278135.

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