Birkhoff-Kakutani theorem

0.1 Birkhoff-Kakutani theorem

Theorem 0.1.

A topological group $\mathrm{(}G\mathrm{,}\mathrm{*}\mathrm{,}e\mathrm{)}$ is metrizable if and only if $G$ is Hausdorff and the identity $e$ of $G$ has a countable neighborhood basis. Here $\mathrm{*}$ 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.

Title	Birkhoff-Kakutani theorem
Canonical name	BirkhoffKakutaniTheorem
Date of creation	2013-03-22 18:24:34
Last modified on	2013-03-22 18:24:34
Owner	bci1 (20947)
Last modified by	bci1 (20947)
Numerical id	17
Author	bci1 (20947)
Entry type	Theorem
Classification	msc 22A22
Classification	msc 22A10
Classification	msc 22A05
Related topic	TopologicalGroup2
Related topic	T2Space
Related topic	HomotopyDoubleGroupoidOfAHausdorffSpace