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.