Editing Topological group (section)

== 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.