Editing Topological group (section)

=== 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}}.