PlanetPhysics/Birkhoff Kakutani Theorem

Birkhoff-Kakutani theorem

\begin{theorem}

A [[../TrivialGroupoid/|topological group]] $(G,.,e)$ is metrizable if and only if $G$ is Hausdorff and the [[../Cod/|identity]] $e$ of $G$ has a countable neighborhood basis. Furthermore, if G is metrizable , then $G$ admits a compatible [[../MetricTensor/|metric]] $d$ which is left-invariant, that is, $d(gx,gy)=d(x,y);$ a right-invariant metric $r$ also exists under these conditions. \end{theorem}