PlanetPhysics/Birkhoff Kakutani Theorem

Birkhoff-Kakutani theorem
\begin{theorem}

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. 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. \end{theorem}