Editing Topological group

{{Short description|Group that is a topological space with continuous group action}}
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[[Image:Real number line.svg|thumb|right|350px|The [[real numbers]] form a topological group under [[addition]]]]

In [[mathematics]], '''topological groups''' are the combination of [[Group (mathematics)|groups]] and [[Topological space|topological spaces]], i.e. they are groups and topological spaces at the same time, such that the [[Continuous function|continuity]] condition for the group operations connects these two structures together and consequently they are not independent from each other.{{sfn|Pontrjagin|1946|p=52}}

Topological groups were studied extensively in the period of 1925 to 1940. [[Alfréd Haar|Haar]] and [[André Weil|Weil]] (respectively in 1933 and 1940) showed that the [[Integral|integrals]] and [[Fourier series]] are special cases of a construct that can be defined on a very wide class of topological groups.{{sfn|Hewitt|Ross|1979|p=1}}

Topological groups, along with [[continuous group action]]s, are used to study continuous [[symmetry|symmetries]], which have many applications, for example, [[Symmetry (physics)|in physics]]. In [[functional analysis]], every [[topological vector space]] is an additive topological group with the additional property that scalar multiplication is continuous; consequently, many results from the theory of topological groups can be applied to functional analysis.

== Formal definition ==

A '''topological group''', {{mvar|G}}, is a [[topological space]] that is also a group such that the group operation (in this case product):
:{{math|⋅ : ''G'' × ''G'' → ''G''}},  {{math|    (''x'', ''y'') ↦ ''xy''}}
and the inversion map:
:{{math|<sup>−1</sup> : ''G'' → ''G''}},  {{math| ''x'' ↦ ''x''<sup>−1</sup>}}
are [[Continuous function|continuous]].<ref group=note>''i.e.'' Continuous means that for any open set {{math|''U'' ⊆ ''G''}}, {{math|''f''<sup>−1</sup>(''U'')}} is open in the domain {{math|dom ''f''}} of {{mvar|f}}.</ref> 
Here {{math|''G'' × ''G''}} is viewed as a topological space with the [[product topology]]. 
Such a topology is said to be '''compatible with the group operations''' and is called a '''group topology'''.

;Checking continuity

The product map is continuous if and only if for any {{math|''x'', ''y'' ∈ ''G''}} and any neighborhood {{mvar|W}} of {{math|''xy''}} in {{mvar|G}}, there exist neighborhoods {{mvar|U}} of {{mvar|x}} and {{mvar|V}} of {{mvar|y}} in {{mvar|G}} such that {{math|''U'' ⋅ ''V'' ⊆ ''W''}}, where {{math|''U'' ⋅ ''V'' :{{=}} {''u'' ⋅ ''v'' : ''u'' ∈ ''U'', ''v'' ∈ ''V''}}}. 
The inversion map is continuous if and only if for any {{math|''x'' ∈ ''G''}} and any neighborhood {{mvar|V}} of {{math|''x''<sup>−1</sup>}} in {{mvar|G}}, there exists a neighborhood {{mvar|U}} of {{mvar|x}} in {{mvar|G}} such that {{math|''U''<sup>−1</sup> ⊆ ''V''}}, where {{math|''U''<sup>−1</sup> :{{=}} { ''u''<sup>−1</sup> : ''u'' ∈ ''U'' }}}.

To show that a topology is compatible with the group operations, it suffices to check that the map
:{{math|''G'' × ''G'' → ''G''}}, {{math|(''x'', ''y'') ↦ ''xy''<sup>−1</sup>}}
is continuous. 
Explicitly, this means that for any {{math|''x'', ''y'' ∈ ''G''}} and any neighborhood {{mvar|W}} in {{mvar|G}} of {{math|''xy''<sup>−1</sup>}}, there exist neighborhoods {{mvar|U}} of {{mvar|x}} and {{mvar|V}} of {{mvar|y}} in {{mvar|G}} such that {{math|''U'' ⋅ (''V''<sup>−1</sup>) ⊆ ''W''}}.

;Additive notation

This definition used notation for multiplicative groups; 
the equivalent for additive groups would be that the following two operations are continuous: 
:{{math|+ : ''G'' × ''G'' → ''G ''}}, {{math|  (''x'', ''y'') ↦ ''x'' + ''y''}}
:{{math|− : ''G'' → ''G ''}}, {{math|''x'' ↦ −''x''}}.

;Hausdorffness

Although not part of this definition, many authors<ref>{{harvnb|Armstrong|1997|p=73}}; {{harvnb|Bredon|1997|p=51}}</ref> require that the topology on {{mvar|G}} be [[Hausdorff space|Hausdorff]]. 
One reason for this is that any topological group can be canonically associated with a Hausdorff topological group by taking an appropriate canonical quotient; 
this however, often still requires working with the original non-Hausdorff topological group.
Other reasons, and some equivalent conditions, are discussed below.

This article will not assume that topological groups are necessarily Hausdorff.

;Category

In the language of [[category theory]], topological groups can be defined concisely as [[group object]]s in the [[category of topological spaces]], in the same way that ordinary groups are group objects in the [[category of sets]]. 
Note that the axioms are given in terms of the maps (binary product, unary inverse, and nullary identity), hence are categorical definitions.

=== Homomorphisms ===

A '''homomorphism''' of topological groups means a continuous [[group homomorphism]] {{math|''G'' → ''H''}}. 
Topological groups, together with their homomorphisms, form a [[category theory|category]]. 
A group homomorphism between topological groups is continuous if and only if it is continuous at ''some'' point.{{sfn | Narici | Beckenstein | 2011 | pp=19-45}}

An '''isomorphism''' of topological groups is a [[group isomorphism]] that is also a [[homeomorphism]] of the underlying topological spaces. 
This is stronger than simply requiring a continuous group isomorphism—the inverse must also be continuous. 
There are examples of topological groups that are isomorphic as ordinary groups but not as topological groups. 
Indeed, any non-discrete topological group is also a topological group when considered with the discrete topology. 
The underlying groups are the same, but as topological groups there is not an isomorphism.

== Examples ==

Every group can be trivially made into a topological group by considering it with the [[discrete topology]]; such groups are called [[discrete group]]s. 
In this sense, the theory of topological groups subsumes that of ordinary groups. 
The [[indiscrete topology]] (i.e. the trivial topology) also makes every group into a topological group.

The [[real number]]s, <math>\mathbb{R}</math> with the usual topology form a topological group under addition. 
[[Euclidean space|Euclidean {{mvar|n}}-space]] {{math|<math>\mathbb{R}</math><sup>''n''</sup>}} is also a topological group under addition, and more generally, every [[topological vector space]] forms an (abelian) topological group. 
Some other examples of [[abelian group|abelian]] topological groups are the [[circle group]] {{math|''S''<sup>1</sup>}}, or the [[torus group|torus]] {{math|(''S''<sup>1</sup>)<sup>''n''</sup>}} for any natural number {{mvar|n}}.

The [[classical group]]s are important examples of non-abelian topological groups. For instance, the [[general linear group]] {{math|GL(''n'',<math>\mathbb{R}</math>)}} of all invertible {{mvar|n}}-by-{{mvar|n}} [[Matrix (mathematics)|matrices]] with real entries can be viewed as a topological group with the topology defined by viewing {{math|GL(''n'',<math>\mathbb{R}</math>)}} as a [[subspace (topology)|subspace]] of Euclidean space {{math|<math>\mathbb{R}</math><sup>''n''×''n''</sup>}}. 
Another classical group is the [[orthogonal group]] {{math|O(''n'')}}, the group of all [[linear map]]s from {{math|<math>\mathbb{R}</math><sup>''n''</sup>}} to itself that preserve the [[Euclidean distance|length]] of all vectors. 
The orthogonal group is [[compact space|compact]] as a topological space. Much of [[Euclidean geometry]] can be viewed as studying the structure of the orthogonal group, or the closely related group {{math|''O''(''n'') ⋉ <math>\mathbb{R}</math><sup>''n''</sup>}} of [[Euclidean group|isometries]] of {{math|<math>\mathbb{R}</math><sup>''n''</sup>}}.

The groups mentioned so far are all [[Lie group]]s, meaning that they are [[smooth manifold]]s in such a way that the group operations are [[smooth function|smooth]], not just continuous. 
Lie groups are the best-understood topological groups; many questions about Lie groups can be converted to purely algebraic questions about [[Lie algebra]]s and then solved.

An example of a topological group that is not a Lie group is the additive group <math>\mathbb{Q}</math> of [[rational number]]s, with the topology inherited from <math>\mathbb{R}</math>. 
This is a [[countable]] space, and it does not have the discrete topology. 
An important example for [[number theory]] is the group {{math|<math>\mathbb{Z}</math><sub>''p''</sub>}} of [[p-adic integers|''p''-adic integers]], for a [[prime number]] {{mvar|p}}, meaning the [[inverse limit]] of the finite groups {{math|<math>\mathbb{Z}</math>/''p''<sup>''n''</sup>}} as ''n'' goes to infinity. 
The group {{math|<math>\mathbb{Z}</math><sub>''p''</sub>}} is well behaved in that it is compact (in fact, homeomorphic to the [[Cantor set]]), but it differs from (real) Lie groups in that it is [[totally disconnected group|totally disconnected]]. 
More generally, there is a theory of [[p-adic Lie group|''p''-adic Lie group]]s, including compact groups such as {{math|GL(''n'',<math>\mathbb{Z}</math><sub>''p''</sub>)}} as well as [[locally compact group]]s such as {{math|GL(''n'',<math>\mathbb{Q}</math><sub>''p''</sub>)}}, where {{math|<math>\mathbb{Q}</math><sub>''p''</sub>}} is the locally compact [[field (mathematics)|field]] of [[p-adic number|''p''-adic number]]s.

The group {{math|<math>\mathbb{Z}</math><sub>''p''</sub>}} is a [[pro-finite group]]; it is isomorphic to a subgroup of the product <math>\prod_{n \geq 1} \mathbb{Z} / p^n </math> in such a way that its topology is induced by the product topology, where the finite groups <math>\mathbb{Z} / p^n</math> are given the discrete topology. 
Another large class of pro-finite groups important in number theory are [[absolute Galois group]]s.

Some topological groups can be viewed as [[infinite dimensional Lie group]]s; this phrase is best understood informally, to include several different families of examples. 
For example, a [[topological vector space]], such as a [[Banach space]] or [[Hilbert space]], is an abelian topological group under addition. Some other infinite-dimensional groups that have been studied, with varying degrees of success, are [[loop group]]s, [[Kac–Moody algebra|Kac–Moody group]]s, [[diffeomorphism#Diffeomorphism group|Diffeomorphism group]]s, [[homeomorphism group]]s, and [[gauge group]]s.

In every [[Banach algebra]] with multiplicative identity, the set of invertible elements forms a topological group under multiplication. 
For example, the group of invertible [[bounded operator]]s on a Hilbert space arises this way.

== Properties ==

=== Translation invariance ===

Every topological group's topology is {{em|{{visible anchor|translation invariant}}}}, which by definition means that if for any <math>a \in G,</math> left or right multiplication by this element yields a homeomorphism <math>G \to G.</math> 
Consequently, for any <math>a \in G</math> and <math>S \subseteq G,</math> the subset <math>S</math> is [[Open set|open]] (resp. [[Closed set|closed]]) in <math>G</math> if and only if this is true of its left translation <math>a S := \{a s : s \in S\}</math> and right translation <math>S a := \{s a : s \in S\}.</math>
If <math>\mathcal{N}</math> is a [[neighborhood basis]] of the identity element in a topological group <math>G</math> then for all <math>x \in X,</math>
<math group=note>x \mathcal{N} := \{x N : N \in \mathcal{N}\}</math>
is a neighborhood basis of <math>x</math> in <math>G.</math>{{sfn|Narici|Beckenstein|2011|pp=19-45}} 
In particular, any group topology on a topological group is completely determined by any neighborhood basis at the identity element. 
If <math>S</math> is any subset of <math>G</math> and <math>U</math> is an open subset of <math>G,</math> then <math>S U := \{s u : s \in S, u \in U\}</math> is an open subset of <math>G.</math>{{sfn|Narici|Beckenstein|2011|pp=19-45}}

=== Symmetric neighborhoods ===

The inversion operation <math>g \mapsto g^{-1}</math> on a topological group <math>G</math> is a homeomorphism from <math>G</math> to itself. 

A subset <math>S \subseteq G</math> is said to be [[Symmetric set|symmetric]] if <math>S^{-1} = S,</math> where <math>S^{-1} := \left\{s^{-1} : s \in S\right\}.</math> 
The closure of every symmetric set in a commutative topological group is symmetric.{{sfn|Narici|Beckenstein|2011|pp=19-45}} 
If {{mvar|S}} is any subset of a commutative topological group {{mvar|G}}, then the following sets are also symmetric: {{math|''S''<sup>−1</sup> ∩ ''S''}}, {{math|''S''<sup>−1</sup> ∪ ''S''}}, and {{math|''S''<sup>−1</sup> ''S''}}.{{sfn|Narici|Beckenstein|2011|pp=19-45}}

For any neighborhood {{mvar|N}} in a commutative topological group {{mvar|G}} of the identity element, there exists a symmetric neighborhood {{mvar|M}} of the identity element such that {{math|''M''<sup>−1</sup> ''M'' ⊆ ''N''}}, where note that {{math|''M''<sup>−1</sup> ''M''}} is necessarily a symmetric neighborhood of the identity element.{{sfn|Narici|Beckenstein|2011|pp=19-45}} 
Thus every topological group has a neighborhood basis at the identity element consisting of symmetric sets.

If {{mvar|G}} is a [[locally compact]] commutative group, then for any neighborhood {{mvar|N}} in {{mvar|G}} of the identity element, there exists a symmetric relatively compact neighborhood {{mvar|M}} of the identity element such that {{math|cl ''M'' ⊆ ''N''}} (where {{math|cl ''M''}} is symmetric as well).{{sfn|Narici|Beckenstein|2011|pp=19-45}}

=== Uniform space ===

Every topological group can be viewed as a [[uniform space]] in two ways; the ''left uniformity'' turns all left multiplications into uniformly continuous maps while the ''right uniformity'' turns all right multiplications into uniformly continuous maps.{{sfn|Bourbaki|1998|loc=section III.3}}
If {{mvar|G}} is not abelian, then these two need not coincide. 
The uniform structures allow one to talk about notions such as [[completeness (topology)|completeness]], [[uniformly continuous|uniform continuity]] and [[uniform convergence]] on topological groups.

=== Separation properties ===

If {{mvar|U}} is an open subset of a commutative topological group {{mvar|G}} and {{mvar|U}} contains a compact set {{mvar|K}}, then there exists a neighborhood {{mvar|N}} of the identity element such that {{math|''KN'' ⊆ ''U''}}.{{sfn|Narici|Beckenstein|2011|pp=19-45}}

As a uniform space, every commutative topological group is [[completely regular space|completely regular]]. 
Consequently, for a multiplicative topological group {{mvar|G}} with identity element 1, the following are equivalent:{{sfn|Narici|Beckenstein|2011|pp=19-45}} 
<ol>
<li>{{mvar|G}} is a T<sub>0</sub>-space ([[Kolmogorov space|Kolmogorov]]);</li>
<li>{{mvar|G}} is a T<sub>2</sub>-space ([[Hausdorff space|Hausdorff]]);</li>
<li>{{mvar|G}} is a T<sub>3{{frac|1|2}}</sub> ([[Tychonoff space|Tychonoff]]);</li>
<li>{{math|{ 1 } }} is closed in {{mvar|G}};</li>
<li>{{math|{ 1 } :{{=}} {{underset|N ∈ 𝒩|{{big|∩}}}} ''N''}}, where {{math|𝒩}} is a neighborhood basis of the identity element in {{mvar|G}};</li>
<lI>for any <math>x \in G</math> such that <math>x \neq 1,</math> there exists a neighborhood {{mvar|U}} in {{mvar|G}} of the identity element such that <math>x \not\in U.</math></li>
</ol>

A subgroup of a commutative topological group is discrete if and only if it has an [[isolated point]].{{sfn|Narici|Beckenstein|2011|pp=19-45}}

If {{mvar|G}} is not Hausdorff, then one can obtain a Hausdorff group by passing to the quotient group {{math|''G''/''K''}}, where {{mvar|K}} is the [[closure (topology)|closure]] of the identity.{{sfn|Bourbaki|1998|loc=section III.2.7}} 
This is equivalent to taking the [[Kolmogorov space#The Kolmogorov quotient|Kolmogorov quotient]] of {{mvar|G}}.

=== Metrisability ===

Let {{mvar|G}} be a topological group. As with any topological space, we say that {{mvar|G}} is [[Metrizable space|metrisable]] if and only if there exists a metric {{mvar|d}} on {{mvar|G}}, which induces the same topology on <math>G</math>. A metric {{mvar|d}} on {{mvar|G}} is called

* ''left-invariant'' (resp. ''right-invariant'') if and only if <math>d(ax_{1},ax_{2})=d(x_{1},x_{2})</math>(resp. <math>d(x_{1}a,x_{2}a)=d(x_{1},x_{2})</math>) for all <math>a,x_{1},x_{2}\in G</math> (equivalently, <math>d</math> is left-invariant just in case the map <math>x \mapsto ax</math> is an [[isometry]] from <math>(G,d)</math> to itself for each <math>a \in G</math>).
* ''proper'' if and only if all open balls, <math>B(r)=\{g \in G \mid d(g,1)<r\}</math> for <math>r>0</math>, are pre-compact.
The '''Birkhoff–Kakutani theorem''' (named after mathematicians [[Garrett Birkhoff]] and [[Shizuo Kakutani]]) states that the following three conditions on a topological group {{mvar|G}} are equivalent:{{sfn|Montgomery|Zippin|1955|loc=section 1.22}}

# {{mvar|G}} is ([[Hausdorff space|Hausdorff]] and) [[First countable space|first countable]] (equivalently: the identity element 1 is closed in {{mvar|G}}, and there is a countable [[neighborhood basis|basis of neighborhoods]] for 1 in {{mvar|G}}).
# {{mvar|G}} is [[metrisable]] (as a topological space).
# There is a left-invariant metric on {{mvar|G}} that induces the given topology on {{mvar|G}}.
# There is a right-invariant metric on {{mvar|G}} that induces the given topology on {{mvar|G}}.

Furthermore, the following are equivalent for any topological group {{mvar|G}}:

# {{mvar|G}} is a [[Second countable space|second countable]] [[Locally compact space|locally compact]] (Hausdorff) space.
# {{mvar|G}} is a [[Polish space|Polish]], [[Locally compact space|locally compact]] (Hausdorff) space.
# {{mvar|G}} is properly [[metrisable]] (as a topological space).
# There is a left-invariant, proper metric on {{mvar|G}} that induces the given topology on {{mvar|G}}.

'''Note:''' As with the rest of the article we of assume here a Hausdorff topology.
The implications 4 <math>\Rightarrow</math> 3 <math>\Rightarrow</math> 2 <math>\Rightarrow</math> 1 hold in any topological space. In particular 3 <math>\Rightarrow</math> 2 holds, since in particular any properly metrisable space is countable union of compact metrisable and thus separable (''cf.'' [[Compact space#Metric spaces|properties of compact metric spaces]]) subsets.
The non-trivial implication 1 <math>\Rightarrow</math> 4 was first proved by Raimond Struble in 1974.<ref>{{Cite journal|last=Struble|first=Raimond A.|date=1974|title=Metrics in locally compact groups|url=http://www.numdam.org/item/?id=CM_1974__28_3_217_0|journal=Compositio Mathematica|language=en|volume=28|issue=3|pages=217–222}}</ref> An alternative approach was made by [[Uffe Haagerup]] and Agata Przybyszewska in 2006,<ref>{{Citation|last1=Haagerup|first1=Uffe|title=Proper metrics on locally compact groups, and proper affine isometric actions on|date=2006|last2=Przybyszewska|first2=Agata|citeseerx=10.1.1.236.827 }}</ref>
the idea of the which is as follows:
One relies on the construction of a left-invariant metric, <math>d_{0}</math>, as in the case of [[First countable space|first countable spaces]]. By local compactness, closed balls of sufficiently small radii are compact, and by normalising we can assume this holds for radius 1. Closing the open ball, {{mvar|U}}, of radius 1 under multiplication yields a [[clopen]] subgroup, {{mvar|H}}, of {{mvar|G}}, on which the metric <math>d_{0}</math> is proper. Since {{mvar|H}} is open and {{mvar|G}} is [[Second countable space|second countable]], the subgroup has at most countably many cosets. One now uses this sequence of cosets and the metric on {{mvar|H}} to construct a proper metric on {{mvar|G}}.

=== Subgroups ===

Every [[subgroup]] of a topological group is itself a topological group when given the [[subspace topology]]. 
Every open subgroup {{mvar|H}} is also closed in {{mvar|G}}, since the complement of {{mvar|H}} is the open set given by the union of open sets {{math|''gH''}} for {{math|''g'' ∈ ''G'' \ ''H''}}. 
If {{mvar|H}} is a subgroup of {{mvar|G}} then the closure of {{mvar|H}} is also a subgroup. 
Likewise, if {{mvar|H}} is a normal subgroup of {{mvar|G}}, the closure of {{mvar|H}} is normal in {{mvar|G}}.

=== Quotients and normal subgroups ===

If {{mvar|H}} is a subgroup of {{mvar|G}}, the set of left [[coset]]s {{math|''G''/''H''}} with the [[quotient topology]] is called a [[homogeneous space]] for {{mvar|G}}. 
The quotient map <math>q : G \to G / H</math> is always [[open map|open]]. 
For example, for a positive integer {{mvar|n}}, the [[n-sphere|sphere]] {{math|''S''<sup>''n''</sup>}} is a homogeneous space for the [[rotation group]] {{math|SO(''n''+1)}} in {{math|<math>\R</math><sup>''n''+1</sup>}}, with {{math|''S''<sup>''n''</sup> {{=}} SO(''n''+1)/SO(''n'')}}. 
A homogeneous space {{math|''G''/''H''}} is Hausdorff if and only if {{mvar|H}} is closed in {{mvar|G}}.{{sfn|Bourbaki|1998|loc=section III.2.5}} 
Partly for this reason, it is natural to concentrate on closed subgroups when studying topological groups.

If {{mvar|H}} is a [[normal subgroup]] of {{mvar|G}}, then the [[quotient group]] {{math|''G''/''H''}} becomes a topological group when given the quotient topology. 
It is Hausdorff if and only if {{mvar|H}} is closed in {{mvar|G}}. 
For example, the quotient group {{math|<math>\R / \Z</math>}} is isomorphic to the circle group {{math|''S''<sup>1</sup>}}.

In any topological group, the [[identity component]] (i.e., the [[connected component (topology)|connected component]] containing the identity element) is a closed normal subgroup. 
If {{mvar|C}} is the identity component and ''a'' is any point of {{mvar|G}}, then the left coset {{math|''aC''}} is the component of {{mvar|G}} containing ''a''. 
So the collection of all left cosets (or right cosets) of {{mvar|C}} in {{mvar|G}} is equal to the collection of all components of {{mvar|G}}. 
It follows that the quotient group {{math|''G''/''C''}} is [[totally disconnected]].{{sfn|Bourbaki|1998|loc=section I.11.5}}

=== Closure and compactness ===

In any commutative topological group, the product (assuming the group is multiplicative) {{math|''KC''}} of a compact set {{mvar|K}} and a closed set {{mvar|C}} is a closed set.{{sfn|Narici|Beckenstein|2011|pp=19-45}} 
Furthermore, for any subsets {{mvar|R}} and {{mvar|S}} of {{mvar|G}}, {{math|(cl ''R'')(cl ''S'') ⊆ cl (''RS'')}}.{{sfn|Narici|Beckenstein|2011|pp=19-45}}

If {{mvar|H}} is a subgroup of a commutative topological group {{mvar|G}} and if {{mvar|N}} is a neighborhood in {{mvar|G}} of the identity element such that {{mvar|''H'' ∩ cl ''N''}} is closed, then {{mvar|H}} is closed.{{sfn|Narici|Beckenstein|2011|pp=19-45}} 
Every discrete subgroup of a Hausdorff commutative topological group is closed.{{sfn|Narici|Beckenstein|2011|pp=19-45}}

=== Isomorphism theorems ===

The [[isomorphism theorem]]s from ordinary group theory are not always true in the topological setting. 
This is because a bijective homomorphism need not be an isomorphism of topological groups. 

For example, a native version of the first isomorphism theorem is false for topological groups: if <math>f:G\to H</math> is a morphism of topological groups (that is, a continuous homomorphism), it is not necessarily true that the induced homomorphism <math>\tilde {f}:G/\ker f\to \mathrm{Im}(f)</math> is an isomorphism of topological groups; it will be a bijective, continuous homomorphism, but it will not necessarily be a homeomorphism. In other words, it will not necessarily admit an inverse in the [[Category theory|category]] 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 <math>f : G \to H</math> is a continuous homomorphism, then the induced homomorphism from {{math|''G''/ker(''f'')}} to {{math|im(''f'')}} is an isomorphism if and only if the map {{mvar|f}} is open onto its image.{{sfn|Bourbaki|1998|loc=section III.2.8}}

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 <math>G \to H</math> is smooth. 
It follows that a topological group has a unique structure of a Lie group if one exists. 
Also, [[closed subgroup theorem|Cartan's theorem]] says that every closed subgroup of a Lie group is a Lie subgroup, in particular a smooth [[submanifold]].

'''[[Hilbert's fifth problem]]''' asked whether a topological group {{mvar|G}} that is a [[topological manifold]] must be a Lie group. 
In other words, does {{mvar|G}} have the structure of a smooth manifold, making the group operations smooth? 
As shown by [[Andrew Gleason]], [[Deane Montgomery]], and [[Leo Zippin]], the answer to this problem is yes.{{sfn|Montgomery|Zippin|1955|loc=section 4.10}} 
In fact, {{mvar|G}} has a [[real analytic]] structure. 
Using the smooth structure, one can define the Lie algebra of {{mvar|G}}, an object of [[linear algebra]] that determines a [[connected space|connected]] group {{mvar|G}} up to [[covering space]]s. 
As a result, the solution to Hilbert's fifth problem reduces the classification of topological groups that are topological manifolds to an algebraic problem, albeit a complicated problem in general.

The theorem also has consequences for broader classes of topological groups. First, every [[compact group]] (understood to be Hausdorff) is an inverse limit of compact Lie groups. 
(One important case is an inverse limit of finite groups, called a [[profinite group]]. For example, the group {{math|<math>\mathbb{Z}</math><sub>''p''</sub>}} of ''p''-adic integers and the [[absolute Galois group]] of a field are profinite groups.) 
Furthermore, every connected locally compact group is an inverse limit of connected Lie groups.{{sfn|Montgomery|Zippin|1955|loc=section 4.6}} 
At the other extreme, a totally disconnected locally compact group always contains a compact open subgroup, which is necessarily a profinite group.{{sfn|Bourbaki|1998|loc=section III.4.6}} 
(For example, the locally compact group {{math|GL(''n'',<math>\mathbb{Q}</math><sub>''p''</sub>)}} contains the compact open subgroup {{math|GL(''n'',<math>\mathbb{Z}</math><sub>''p''</sub>)}}, which is the inverse limit of the finite groups {{math|GL(''n'',<math>\mathbb{Z}</math>/''p''<sup>''r''</sup>)}} as {{mvar|r}}' goes to infinity.)

== Representations of compact or locally compact groups ==

An '''action''' of a topological group {{mvar|G}} on a topological space ''X'' is a [[Group action (mathematics)|group action]] of {{mvar|G}} on ''X'' such that the corresponding function {{math|''G'' × ''X'' → ''X''}} is continuous. 
Likewise, a '''[[representation theory|representation]]''' of a topological group {{mvar|G}} on a real or complex topological vector space ''V'' is a continuous action of {{mvar|G}} on ''V'' such that for each {{math|''g'' ∈ ''G''}}, the map {{math|''v'' ↦ ''gv''}} from ''V'' to itself is linear.

Group actions and representation theory are particularly well understood for compact groups, generalizing what happens for [[representation theory of finite groups|finite group]]s. 
For example, every finite-dimensional (real or complex) representation of a compact group is a [[direct sum]] of [[irreducible representation]]s. 
An infinite-dimensional [[unitary representation]] of a compact group can be decomposed as a Hilbert-space direct sum of irreducible representations, which are all finite-dimensional; this is part of the [[Peter–Weyl theorem]].{{sfn|Hewitt|Ross|1970|loc=Theorem 27.40}} 
For example, the theory of [[Fourier series]] describes the decomposition of the unitary representation of the circle group {{math|''S''<sup>1</sup>}} on the complex Hilbert space {{math|L<sup>2</sup>(''S''<sup>1</sup>)}}. 
The irreducible representations of {{math|''S''<sup>1</sup>}} are all 1-dimensional, of the form {{math|''z'' ↦ ''z''<sup>''n''</sup>}} for integers {{mvar|n}} (where {{math|''S''<sup>1</sup>}} is viewed as a subgroup of the multiplicative group {{mvar|<math>\mathbb{C}</math>}}*). 
Each of these representations occurs with multiplicity 1 in {{math|L<sup>2</sup>(''S''<sup>1</sup>)}}.

The irreducible representations of all compact connected Lie groups have been classified. 
In particular, the [[character theory|character]] of each irreducible representation is given by the [[Weyl character formula]].

More generally, locally compact groups have a rich theory of [[harmonic analysis]], because they admit a natural notion of [[measure (mathematics)|measure]] and [[integral]], given by the [[Haar measure]]. 
Every unitary representation of a locally compact group can be described as a [[direct integral]] of irreducible unitary representations. 
(The decomposition is essentially unique if {{mvar|G}} is of [[C*-algebra#Type for C*-algebras|Type I]], which includes the most important examples such as abelian groups and [[semisimple Lie group]]s.{{sfn|Mackey|1976|loc=section 2.4}}) 
A basic example is the [[Fourier transform]], which decomposes the action of the additive group <math>\mathbb{R}</math> on the Hilbert space {{math|L<sup>2</sup>(<math>\mathbb{R}</math>)}} as a direct integral of the irreducible unitary representations of <math>\mathbb{R}</math>. 
The irreducible unitary representations of <math>\mathbb{R}</math> are all 1-dimensional, of the form {{math|''x'' ↦ ''e''<sup>2π''iax''</sup>}} for {{math|''a'' ∈ <math>\mathbb{R}</math>}}.

The irreducible unitary representations of a locally compact group may be infinite-dimensional. 
A major goal of representation theory, related to the [[Langlands classification]] of [[admissible representation]]s, is to find the [[unitary dual]] (the space of all irreducible unitary representations) for the semisimple Lie groups. 
The unitary dual is known in many cases such as {{math|[[representation theory of SL2(R)|SL(2,<math>\mathbb{R}</math>)]]}}, but not all.

For a [[locally compact abelian group]] {{mvar|G}}, every irreducible unitary representation has dimension 1. 
In this case, the unitary dual <math>\hat{G}</math> is a group, in fact another locally compact abelian group. 
[[Pontryagin duality]] states that for a locally compact abelian group {{mvar|G}}, the dual of <math>\hat{G}</math> is the original group {{mvar|G}}. 
For example, the dual group of the integers {{math|<math>\mathbb{Z}</math>}} is the circle group {{math|''S''<sup>1</sup>}}, while the group <math>\mathbb{R}</math> of real numbers is isomorphic to its own dual.

Every locally compact group {{mvar|G}} has a good supply of irreducible unitary representations; for example, enough representations to distinguish the points of {{mvar|G}} (the [[Gelfand–Raikov theorem]]). 
By contrast, representation theory for topological groups that are not locally compact has so far been developed only in special situations, and it may not be reasonable to expect a general theory. 
For example, there are many abelian [[Banach manifold|Banach–Lie groups]] for which every representation on Hilbert space is trivial.{{sfn|Banaszczyk|1983}}

== Homotopy theory of topological groups ==

Topological groups are special among all topological spaces, even in terms of their [[homotopy type]]. 
One basic point is that a topological group {{mvar|G}} determines a path-connected topological space, the [[classifying space]] {{math|''BG''}} (which classifies [[principal bundle|principal {{mvar|G}}-bundles]] over topological spaces, under mild hypotheses). 
The group {{mvar|G}} is isomorphic in the [[model category#Homotopy and the homotopy category|homotopy category]] to the [[loop space]] of {{math|''BG''}}; that implies various restrictions on the homotopy type of {{mvar|G}}.{{sfn|Hatcher|2001|loc=Theorem 4.66}} 
Some of these restrictions hold in the broader context of [[H-space]]s.

For example, the [[fundamental group]] of a topological group {{mvar|G}} is abelian. 
(More generally, the [[Whitehead product]] on the homotopy groups of {{mvar|G}} is zero.) 
Also, for any field ''k'', the [[cohomology]] ring {{math|''H''*(''G'',''k'')}} has the structure of a [[Hopf algebra]]. 
In view of structure theorems on Hopf algebras by [[Heinz Hopf]] and [[Armand Borel]], this puts strong restrictions on the possible cohomology rings of topological groups. 
In particular, if {{mvar|G}} is a path-connected topological group whose rational cohomology ring {{math|''H''*(''G'',<math>\mathbb{Q}</math>)}} is finite-dimensional in each degree, then this ring must be a free [[graded-commutative]] algebra over <math>\mathbb{Q}</math>, that is, the [[tensor product of algebras|tensor product]] of a [[polynomial ring]] on generators of even degree with an [[exterior algebra]] on generators of odd degree.{{sfn|Hatcher|2001|loc=Theorem 3C.4}}

In particular, for a connected Lie group {{mvar|G}}, the rational cohomology ring of {{mvar|G}} is an exterior algebra on generators of odd degree. 
Moreover, a connected Lie group {{mvar|G}} has a [[maximal compact subgroup]] ''K'', which is unique up to conjugation, and the inclusion of ''K'' into {{mvar|G}} is a [[homotopy equivalence]]. 
So describing the homotopy types of Lie groups reduces to the case of compact Lie groups. 
For example, the maximal compact subgroup of {{math|SL(2,<math>\mathbb{R}</math>)}} is the circle group {{math|SO(2)}}, and the homogeneous space {{math|SL(2,<math>\mathbb{R}</math>)/SO(2)}} can be identified with the [[hyperbolic plane]]. 
Since the hyperbolic plane is [[contractible space|contractible]], the inclusion of the circle group into {{math|SL(2,<math>\mathbb{R}</math>)}} is a homotopy equivalence.

Finally, compact connected Lie groups have been classified by [[Wilhelm Killing]], [[Élie Cartan]], and [[Hermann Weyl]]. 
As a result, there is an essentially complete description of the possible homotopy types of Lie groups. 
For example, a compact connected Lie group of dimension at most 3 is either a torus, the group [[SU(2)]] ([[diffeomorphic]] to the 3-sphere {{math|''S''<sup>3</sup>}}), or its quotient group {{math|SU(2)/{±1} ≅ [[SO(3)]]}} (diffeomorphic to {{math|[[real projective space|'''RP'''<sup>3</sup>]]}}).

==Complete topological group{{anchor|Complete abelian topological groups}}==

{{See also|Complete uniform space}}

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

===Canonical uniformity on a commutative topological group===

{{Main|Uniform space}}

This article will henceforth assume that any topological group that we consider is an additive commutative topological group with identity element <math>0.</math>

The '''diagonal''' of <math>X</math> is the set
<math display=block>\Delta_X := \{(x, x) : x \in X\}</math>
and for any <math>N \subseteq X</math> containing <math>0,</math> the '''canonical entourage''' or '''canonical vicinities around <math>N</math>''' is the set
<math display=block>\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\})</math>

For a topological group <math>(X, \tau),</math> the '''canonical uniformity'''{{sfn|Edwards|1995|p=61}} on <math>X</math> is the [[Uniform space|uniform structure]] induced by the set of all canonical entourages <math>\Delta(N)</math> as <math>N</math> ranges over all neighborhoods of <math>0</math> in <math>X.</math>

That is, it is the upward closure of the following prefilter on <math>X \times X,</math> 
<math display=block>\left\{\Delta(N) : N \text{ is a neighborhood of } 0 \text{ in } X\right\}</math>
where this prefilter forms what is known as a [[base of entourages]] of the canonical uniformity. 

For a commutative additive group <math>X,</math> a fundamental system of entourages <math>\mathcal{B}</math> is called a '''translation-invariant uniformity''' if for every <math>B \in \mathcal{B},</math> <math>(x, y) \in B</math> if and only if <math>(x + z, y + z) \in B</math> for all <math>x, y, z \in X.</math> A uniformity <math>\mathcal{B}</math> is called '''translation-invariant''' if it has a base of entourages that is translation-invariant.{{sfn|Schaefer|Wolff|1999|pp=12-19}}

<ul>
<li>The canonical uniformity on any commutative topological group is translation-invariant.</li>
<li>The same canonical uniformity would result by using a neighborhood basis of the origin rather the filter of all neighborhoods of the origin.</li>
<li>Every entourage <math>\Delta_X(N)</math> contains the diagonal <math>\Delta_X := \Delta_X(\{0\}) = \{(x, x) : x \in X\}</math> because <math>0 \in N.</math>
</li>
<li>If <math>N</math> is [[Symmetric set|symmetric]] (that is, <math>-N = N</math>) then <math>\Delta_X(N)</math> is symmetric (meaning that <math>\Delta_X(N)^{\operatorname{op}} = \Delta_X(N)</math>) and
<math>\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).</math>
<li>The topology induced on <math>X</math> by the canonical uniformity is the same as the topology that <math>X</math> started with (that is, it is <math>\tau</math>).</li>
</ul>

===Cauchy prefilters and nets===

{{Main|Filters in topology|Net (mathematics)}}

The general theory of [[uniform space]]s has its own definition of a "Cauchy prefilter" and "Cauchy net." For the canonical uniformity on <math>X,</math> these reduces down to the definition described below.

Suppose <math>x_{\bull} = \left(x_i\right)_{i \in I}</math> is a net in <math>X</math> and <math>y_{\bull} = \left(y_j\right)_{j \in J}</math> is a net in <math>Y.</math> Make <math>I \times J</math> into a directed set by declaring <math>(i, j) \leq \left(i_2, j_2\right)</math> if and only if <math>i \leq i_2 \text{ and } j \leq j_2.</math> Then{{sfn|Narici|Beckenstein|2011|pp=47-66}} <math>x_{\bull} \times y_{\bull}: = \left(x_i, y_j\right)_{(i, j) \in I \times J}</math> denotes the '''product net'''. If <math>X = Y</math> then the image of this net under the addition map <math>X \times X \to X</math> denotes the '''sum''' of these two nets:
<math display=block>x_{\bull} + y_{\bull}: = \left(x_i + y_j\right)_{(i, j) \in I \times J}</math>
and similarly their '''difference''' is defined to be the image of the product net under the subtraction map:
<math display=block>x_{\bull} - y_{\bull}: = \left(x_i - y_j\right)_{(i, j) \in I \times J}.</math>

A [[Net (mathematics)|net]] <math>x_{\bull} = \left(x_i\right)_{i \in I}</math> in an additive topological group <math>X</math> is called a '''Cauchy net''' if{{sfn|Narici|Beckenstein|2011|p=48}}
<math display=block>\left(x_i - x_j\right)_{(i, j) \in I \times I} \to 0 \text{ in } X</math>
or equivalently, if for every neighborhood <math>N</math> of <math>0</math> in <math>X,</math> there exists some <math>i_0 \in I</math> such that 
<math>x_i - x_j \in N</math> for all indices <math>i, j \geq i_0.</math> 

A '''[[Cauchy sequence]]''' is a Cauchy net that is a sequence.

If <math>B</math> is a subset of an additive group <math>X</math> and <math>N</math> is a set containing <math>0,</math> then<math>B</math> is said to be an '''<math>N</math>-small set''' or '''small of order <math>N</math>''' if <math>B - B \subseteq N.</math>{{sfn|Narici|Beckenstein|2011|pp=48-51}}

A prefilter <math>\mathcal{B}</math> on an additive topological group <math>X</math> called a '''Cauchy prefilter''' if it satisfies any of the following equivalent conditions: 
<ol>
<li><math>\mathcal{B} - \mathcal{B} \to 0</math> in <math>X,</math> where <math>\mathcal{B} - \mathcal{B} := \{B - C : B, C \in \mathcal{B}\}</math> is a prefilter.</li>
<li><math>\{B - B : B \in \mathcal{B}\} \to 0</math> in <math>X,</math> where <math>\{B - B : B \in \mathcal{B}\}</math> is a prefilter equivalent to <math>\mathcal{B} - \mathcal{B}.</math></li>
<li>For every neighborhood <math>N</math> of <math>0</math> in <math>X,</math> <math>\mathcal{B}</math> contains some <math>N</math>-small set (that is, there exists some <math>B \in \mathcal{B}</math> such that <math>B - B \subseteq N</math>).{{sfn|Narici|Beckenstein|2011|pp=48–51}}</li>
</ol>
and if <math>X</math> is commutative then also:
<ol start=4>
<li>For every neighborhood <math>N</math> of <math>0</math> in <math>X,</math> there exists some <math>B \in \mathcal{B}</math> and some <math>x \in X</math> such that <math>B \subseteq x + N.</math>{{sfn|Narici|Beckenstein|2011|pp=48-51}}</li>
</ol>
* It suffices to check any of the above condition for any given [[neighborhood basis]] of <math>0</math> in <math>X.</math>

Suppose <math>\mathcal{B}</math> is a prefilter on a commutative topological group <math>X</math> and <math>x \in X.</math> Then <math>\mathcal{B} \to x</math> in <math>X</math> if and only if <math>x \in \operatorname{cl} \mathcal{B}</math> and <math>\mathcal{B}</math> is Cauchy.{{sfn|Narici|Beckenstein|2011|pp=47-66}}

===Complete commutative topological group===

{{Main|Complete uniform space}}

Recall that for any <math>S \subseteq X,</math> a prefilter <math>\mathcal{C}</math> '''''on <math>S</math>''''' is necessarily a subset of <math>\wp(S)</math>; that is, <math>\mathcal{C} \subseteq \wp(S).</math>

A subset <math>S</math> of a topological group <math>X</math> is called a '''complete subset''' if it satisfies any of the following equivalent conditions: 
<ol>
<li>Every Cauchy prefilter <math>\mathcal{C} \subseteq \wp(S)</math> on <math>S</math> [[Filters in topology|converges]] to at least one point of <math>S.</math>
* If <math>X</math> is Hausdorff then every prefilter on <math>S</math> will converge to at most one point of <math>X.</math> But if <math>X</math> is not Hausdorff then a prefilter may converge to multiple points in <math>X.</math> The same is true for nets.</li>
<li>Every Cauchy net in <math>S</math> converges to at least one point of <math>S</math>;</li>
<li>Every Cauchy filter <math>\mathcal{C}</math> on <math>S</math> converges to at least one point of <math>S.</math></li>
<li><math>S</math> is a [[Complete uniform space|complete]] uniform space (under the point-set topology definition of "[[complete uniform space]]") when <math>S</math> is endowed with the uniformity induced on it by the canonical uniformity of <math>X</math>;</li>
</ol>

A subset <math>S</math> is called a '''sequentially complete subset''' if every Cauchy sequence in <math>S</math> (or equivalently, every elementary Cauchy filter/prefilter on <math>S</math>) converges to at least one point of <math>S.</math>

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

A commutative topological group <math>X</math> is called a '''complete group''' if any of the following equivalent conditions hold:
<ol>
<li><math>X</math> is complete as a subset of itself.</li>
<li>Every Cauchy net in <math>X</math> [[Net (mathematics)|converges]] to at least one point of <math>X.</math></li>
<li>There exists a neighborhood of <math>0</math> in <math>X</math> that is also a complete subset of <math>X.</math>{{sfn|Narici|Beckenstein|2011|pp=48-51}}
* This implies that every locally compact commutative topological group is complete.</li>
<li>When endowed with its canonical uniformity, <math>X</math> becomes is a [[complete uniform space]].
* In the general theory of [[uniform space]]s, a uniform space is called a [[complete uniform space]] if each Cauchy [[Net (mathematics)|filter]] in <math>X</math> converges in <math>(X, \tau)</math> to some point of <math>X.</math></li>
</ol>

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

'''Neighborhood basis''': Suppose <math>C</math> is a completion of a commutative topological group <math>X</math> with <math>X \subseteq C</math> and that <math>\mathcal{N}</math> is a [[neighborhood base]] of the origin in <math>X.</math> Then the family of sets 
<math display=block>\left\{ \operatorname{cl}_C N : N \in \mathcal{N} \right\}</math>
is a neighborhood basis at the origin in <math>C.</math>{{sfn|Narici|Beckenstein|2011|pp=47-66}}

'''{{visible anchor|Uniform continuity}}'''

Let <math>X</math> and <math>Y</math> be topological groups, <math>D \subseteq X,</math> and <math>f : D \to Y</math> be a map. Then <math>f : D \to Y</math> is '''uniformly continuous''' if for every neighborhood <math>U</math> of the origin in <math>X,</math> there exists a neighborhood <math>V</math> of the origin in <math>Y</math> such that for all <math>x, y \in D,</math> if <math>y - x \in U</math> then <math>f(y) - f(x) \in V.</math>

== Generalizations ==

Various generalizations of topological groups can be obtained by weakening the continuity conditions:{{sfn|Arhangel'skii|Tkachenko|2008|p=12}}
* A [[semitopological group]] is a group {{mvar|G}} with a topology such that for each {{math|''c'' ∈ ''G''}} the two functions {{math|''G'' → ''G''}} defined by {{math|''x'' ↦ ''xc''}} and {{math|''x'' ↦ ''cx''}} are continuous.
* A [[quasitopological group]] is a semitopological group in which the function mapping elements to their inverses is also continuous.
* A [[paratopological group]] is a group with a topology such that the group operation is continuous.

==See also==

* {{annotated link|Algebraic group}}
* {{annotated link|Complete field}}
* {{annotated link|Compact group}}
* {{annotated link|Complete topological vector space}}
* {{annotated link|Lie group}}
* {{annotated link|Locally compact field}}
* {{annotated link|Locally compact group}}
* {{annotated link|Locally compact quantum group}}
* {{annotated link|Profinite group}}
* {{annotated link|Ordered topological vector space}}
* {{annotated link|Topological abelian group}}
* {{annotated link|Topological field}}
* {{annotated link|Topological module}}
* {{annotated link|Topological ring}}
* {{annotated link|Topological semigroup}}
* {{annotated link|Topological vector space}}

== Notes ==
{{reflist|group=note}}

== Citations ==
{{reflist|30em}}

== References ==
{{refbegin|2}}
* {{cite book | last1=Arhangel'skii|first1= Alexander | author-link1 = Alexander Arhangelskii | last2=Tkachenko|first2= Mikhail | title=Topological Groups and Related Structures | publisher=[[World Scientific]] | year=2008 | isbn=978-90-78677-06-2 | mr=2433295}}
* {{cite book | last=Armstrong|first= Mark A. | title=Basic Topology | publisher=[[Springer-Verlag]] | edition=1st | year=1997 | isbn=0-387-90839-0 | mr=0705632}}
* {{Citation | author1-first=Wojciech | author1-last=Banaszczyk | title=On the existence of exotic Banach–Lie groups | journal=[[Mathematische Annalen]] | volume=264 | issue=4 | year=1983 | pages=485–493 | doi=10.1007/BF01456956 | mr=0716262 | s2cid=119755117 | url=https://polipapers.upv.es/index.php/AGT/article/view/1800 }}
* {{Citation | author1-first=Nicolas | author1-last=Bourbaki | author1-link=Nicolas Bourbaki | year=1998 | title=General Topology. Chapters 1–4 | publisher=[[Springer-Verlag]] | isbn=3-540-64241-2 | mr=1726779}}
* {{Citation | last=Folland| first=Gerald B. | author1-link=Gerald Folland | title=A Course in Abstract Harmonic Analysis| year=1995 | publisher=[[CRC Press]] | isbn=0-8493-8490-7}}
* {{cite book | author-link = Glen Bredon | last=Bredon|first= Glen E. | title=Topology and Geometry | series=Graduate Texts in Mathematics | publisher=[[Springer-Verlag]] | edition=1st | year=1997 | isbn=0-387-97926-3 | mr=1700700}}
* {{Citation | author1-first=Allen | author1-last=Hatcher | author1-link=Allen Hatcher | title=Algebraic Topology | publisher=[[Cambridge University Press]] | year=2001 | url=https://pi.math.cornell.edu/~hatcher/AT/ATpage.html | isbn=0-521-79540-0 | mr=1867354}}
* {{Edwards Functional Analysis Theory and Applications}}
* {{Citation | author1-last=Hewitt | author1-first=Edwin | author1-link=Edwin Hewitt | author2-first=Kenneth A. | author2-last=Ross | author2-link=Kenneth A. Ross | title=Abstract Harmonic Analysis | year=1979 | volume=1 | edition=2nd | publisher=[[Springer-Verlag]] | isbn=978-0387941905 | mr=0551496}}
* {{Citation | author1-last=Hewitt | author1-first=Edwin | author1-link=Edwin Hewitt | author2-first=Kenneth A. | author2-last=Ross | author2-link=Kenneth A. Ross | title=Abstract Harmonic Analysis | year=1970 | volume=2 | publisher=[[Springer-Verlag]] | isbn=978-0387048321 | mr=0262773}}
* {{Citation | author1-first=George W. | author1-last=Mackey | author1-link=George Mackey | title=The Theory of Unitary Group Representations | publisher=[[University of Chicago Press]] | year=1976 | isbn=0-226-50051-9 | mr=0396826}}
* {{Citation | author1-first=Deane | author1-last=Montgomery | author1-link=Deane Montgomery | author2-first=Leo | author2-last=Zippin | author2-link=Leo Zippin | title=Topological Transformation Groups | mr=0073104 | publisher=[[Interscience Publishers]] | location=New York, London |year=1955}}
* {{Narici Beckenstein Topological Vector Spaces|edition=2}} <!-- {{sfn | Narici | Beckenstein | 2011 | p=}} -->
* {{cite book|author-last=Pontrjagin|author-first=Leon|title=Topological Groups|date=1946|publisher=[[Princeton University Press]]}}
* {{cite book | last = Pontryagin | first = Lev S. | author-link = Lev Pontryagin | title = Topological Groups | year = 1986 | edition = 3rd | others = trans. from Russian by Arlen Brown and P.S.V. Naidu | publisher = Gordon and Breach Science Publishers | location = New York | isbn = 2-88124-133-6 | mr=0201557}}
* {{cite book | last=Porteous | first=Ian R. | author-link=Ian R. Porteous | title=Topological Geometry | publisher=[[Cambridge University Press]] | year=1981 | edition=2nd | isbn=0-521-23160-4 | mr=0606198}}
* {{Schaefer Wolff Topological Vector Spaces|edition=2}}
{{refend}}

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{{DEFAULTSORT:Topological Group}}
[[Category:Topological groups| ]]
[[Category:Lie groups]]
[[Category:Fourier analysis]]