Birkhoff-Kakutani theorem

0.1 Birkhoff-Kakutani theorem

Theorem 0.1.

A topological group $(G,*,e)$ is metrizable if and only if $G$ is Hausdorff and the identity $e$ of $G$ has a countable neighborhood basis. Here $*$ is the group composition law or operation. Furthermore, if G is metrizable, then $G$ admits a compatible metric $d$ which is left-invariant, that is,

 $d(gx,gy)=d(x,y);$

a right-invariant metric $r$ also exists under these conditions.

References

Title Birkhoff-Kakutani theorem BirkhoffKakutaniTheorem 2013-03-22 18:24:34 2013-03-22 18:24:34 bci1 (20947) bci1 (20947) 17 bci1 (20947) Theorem msc 22A22 msc 22A10 msc 22A05 TopologicalGroup2 T2Space HomotopyDoubleGroupoidOfAHausdorffSpace