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{{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. 

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]]