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In [[mathematics]], a '''topological group''' is a [[group (mathematics)|group]] ''G'' together with a [[topological space|topology]] on ''G'' such that the group's binary operation and the group's inverse function are [[continuous function (topology)|continuous]] functions with respect to the topology.<ref>{{cite book |authors=Waerden, Bartel Leendert et al |chapter=Topological algebra |title=Algebra |volume=Vol. 2 |publisher=Springer |year=2003 |isbn=978-0-387-40625-1 |page=256 |url=http://books.google.com/books?id=cvOgYpQ7KMIC&pg=PA256}}</ref> A topological group is a mathematical object with both an algebraic structure and a topological structure. Thus, one may perform algebraic operations, because of the group structure, and one may talk about continuous functions, because of the topology.

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]].

== Formal definition ==
A '''topological group''' ''G'' is a [[topological space]] and [[group (mathematics)|group]] such that the group operations of product:
:<math>G\times G \to G : (x,y)\mapsto xy</math>
and taking inverses:
:<math>G\to G : x \mapsto x^{-1}</math>
are [[continuous function (topology)|continuous function]]s. Here, ''G'' &times; ''G'' is viewed as a topological space by using the [[product topology]].

Although not part of this definition, many authors<ref>Armstrong, p. 73; Bredon, p. 51; Willard, p. 91.</ref> require that the topology on ''G'' be [[Hausdorff space|Hausdorff]]; this corresponds to the identity map <math>* \to G</math> being a [[closed inclusion]] (hence also a [[cofibration]]). The reasons, and some equivalent conditions, are discussed below. In the end, this is not a serious restriction&mdash;any topological group can be made Hausdorff in a canonical fashion.<ref>D. Ramakrishnan and R. Valenza (1999).  "Fourier Analysis on Number Fields".  Springer-Verlag, Graduate Texts in Mathematics.  Pp. 6–7.</ref>

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. Adding the further requirement of Hausdorff (and cofibration) corresponds to refining to a [[model category]].

=== Homomorphisms ===

A [[homomorphism]] between two topological groups ''G'' and ''H'' is just a continuous [[group homomorphism]] ''G'' <math>\to</math> ''H''. An [[isomorphism]] of topological groups is a [[group isomorphism]] which is also a [[homeomorphism]] of the underlying topological spaces. This is stronger than simply requiring a continuous group isomorphism&mdash;the inverse must also be continuous. There are examples of topological groups which are isomorphic as ordinary groups but not as topological groups.  Indeed, any nondiscrete 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.

Topological groups, together with their homomorphisms, form a [[category theory|category]].

== 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 [[real number]]s '''R''', together with addition as operation and its usual topology, form a topological group. More generally, [[Euclidean space|Euclidean ''n''-space]] '''R'''<sup>''n''</sup> with addition and standard topology is a topological group. More generally yet, the additive groups of all [[topological vector space]]s, such as [[Banach space]]s or [[Hilbert space]]s, are topological groups.

The above examples are all [[abelian group|abelian]].  Examples of non-abelian topological groups are given by the [[classical group]]s. For instance, the [[general linear group]] GL(''n'','''R''') of all invertible ''n''-by-''n'' [[Matrix (mathematics)|matrices]] with real entries can be viewed as a topological group with the topology defined by viewing GL(''n'','''R''') as a [[subspace (topology)|subset]] of Euclidean space '''R'''<sup>''n''&times;''n''</sup>. 

An example of a topological group which is not a [[Lie group]] is given by the [[rational number]]s '''Q''' with the topology inherited from '''R'''. This is a [[countable]] space and it does not have the [[discrete topology]]. For a nonabelian example, consider the subgroup of rotations of '''R'''<sup>3</sup> generated by two rotations by irrational multiples of 2π about different axes. 

In every [[Banach algebra]] with multiplicative identity, the set of invertible elements forms a topological group under multiplication.

== Properties ==
The algebraic and topological structures of a topological group interact in non-trivial ways. For example, in any topological group the [[identity component]] (i.e. the [[Locally connected space|connected component]] containing the identity element) is a [[closed set|closed]] [[normal subgroup]]. This is because if ''C'' is the identity component, ''a*C'' is the component of ''G'' (the group) containing a. In fact, the collection of all left cosets (or right cosets) of ''C'' in ''G'' is equal to the collection of all components of ''G''. Therefore, the [[quotient topology]] induced by the quotient map from ''G'' to ''G''/''C'' is [[totally disconnected group|totally disconnected]].<ref>{{springer|id=T/t093070|title=Topological group|author=O.V. Mel'nikov}}</ref>

The inversion operation on a topological group ''G'' is a [[homeomorphism]] from ''G'' to itself. Likewise, if ''a'' is any element of ''G'', then left or right multiplication by ''a'' yields a homeomorphism ''G'' → ''G''.

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. If ''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. 

As a uniform space, every topological group is [[completely regular space|completely regular]]. It follows that if a topological group is T<sub>0</sub> ([[Kolmogorov space|Kolmogorov]]) then it is already T<sub>2</sub> ([[Hausdorff space|Hausdorff]]), even T<sub>3½</sub> ([[Tychonoff space|Tychonoff]]). 

Every [[subgroup]] of a topological group is itself a topological group when given the [[subspace topology]]. If ''H'' is a subgroup of ''G'', the set of left or right [[coset]]s ''G''/''H'' is a topological space when given the [[quotient topology]] (the [[finer topology|finest topology]] on ''G''/''H'' which makes the natural projection ''q'' : ''G'' → ''G''/''H'' continuous). One can show that the quotient map ''q'' : ''G'' → ''G''/''H'' is always [[open map|open]]. 

Every open subgroup ''H'' is also closed, since the complement of ''H'' is the open set given by the union of open sets ''gH'' for ''g'' in G \ H.

If ''H'' is a [[normal subgroup]] of ''G'', then the [[factor group]], ''G''/''H'' becomes a topological group when given the [[quotient topology]]. However, if ''H'' is not [[closed set|closed]] in the topology of ''G'', then ''G''/''H'' will not be T<sub>0</sub> even if ''G'' is. It is therefore natural to restrict oneself to the category of T<sub>0</sub> topological groups, and restrict the definition of ''normal'' to ''normal and closed''.

The [[isomorphism theorem]]s known 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. The theorems are valid if one places certain restrictions on the maps involved. For example, the first isomorphism theorem states that if ''f'' : ''G'' → ''H'' is a homomorphism then ''G''/ker(''f'') is isomorphic to im(''f'') if and only if the map ''f'' is [[open map|open]] onto its image.

If ''H'' is a subgroup of ''G'' then the [[closure (topology)|closure]] of ''H'' is also a subgroup. Likewise, if ''H'' is a normal subgroup, the closure of ''H'' is normal.

A topological group ''G'' is Hausdorff if and only if the trivial one-element subgroup is closed in ''G''. If ''G'' is not Hausdorff then one can obtain a Hausdorff group by passing to the quotient space ''G''/''K'' where ''K'' is the closure of the identity. This is equivalent to taking the [[Kolmogorov space#The Kolmogorov quotient|Kolmogorov quotient]] of ''G''.

The [[fundamental group]] of a topological group is always [[abelian group|abelian]]. This is a special case of the fact that the fundamental group of an H-space is abelian, since topological groups are [[H-space]]s.

==Relationship to other areas of mathematics==

Of particular importance in [[harmonic analysis]] are the [[locally compact group]]s, because they admit a natural notion of [[measure (mathematics)|measure]] and [[integral]], given by the [[Haar measure]]. The theory of [[group representation]]s is almost identical for finite groups and for compact topological groups. In general, [[σ-compact space|σ-compact]] [[Baire space|Baire]] topological groups are locally compact.

{{Expand section|date=September 2008}}

==Generalizations==

Various generalizations of topological groups can be obtained by weakening the continuity conditions:<ref>Arhangel'skii & Tkachenko, p12</ref>
* A ''semitopological group'' is a group ''G'' with a topology such that for each ''c'' in ''G'' the two functions ''G'' → ''G'' defined by <math>x\mapsto xc</math> and <math>x\mapsto cx</math> 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==

*[[Lie group]]
*[[algebraic group]]
*[[profinite group]]
*[[topological ring]]

==Notes==
{{reflist}}

==References==
*{{cite book | author1=Arhangel'skii, Alexander | authorlink1 = Alexander Arhangelskii | author2=Tkachenko, Mikhail | title=Topological Groups and Related Structures | publisher=Atlantis Press | year=2008 | isbn=90-78677-06-6}}
*{{cite book | author=Armstrong, M. A. | title=Basic Topology | publisher=Springer Verlag | edition=1st | year=1997 | isbn=0-387-90839-0 }}
*{{cite book | authorlink = Glen Bredon | author=Bredon, Glen E. | title=Topology and Geometry | series=Graduate Texts in Mathematics | publisher=Springer | edition=1 | year=1997 | isbn=0-387-97926-3}}
*{{cite book | last = Husain | first = Taqdir | title = Introduction to Topological Groups | year = 1981 | publisher = R.E. Krieger Pub. Co | location = Philadelphia | isbn = 0-89874-193-9}}
*{{cite book | last = Pontryagin | first = Lev S. | authorlink = Lev Semenovich 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}}
*{{cite book | last=Porteous | first=I.R. | authorlink=Ian R. Porteous | title=Topological Geometry | publisher=Van Nostrand Reinhold | isbn=0-442-06606-6 | year=1969 | isbn=0-442-06606-6 | zbl=0186.06304  | pages=336–352 }}
*{{cite book | author=Willard, Stephen | title=General Topology | publisher=Dover Publications | year=2004 | isbn=0-486-43479-6}}

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[[Category:Topological groups| ]]
[[Category:Fourier analysis]]