Editing Topological group (section)

== 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.