= Birkhoff-Kakutani theorem
{c}
{title2=$G\text{ metrizable}\iff G\text{ first countable}$}
A Hausdorff <topological group> admits a compatible <left-invariant group metric> exactly when it has a countable <neighbourhood basis> at the <identity element>. Choose symmetric <neighbourhoods> of the <identity element> $U_n$ with $U_{n+1}^3\subset U_n$ and trivial intersection. Assign a step in $U_n$ cost $2^{-n}$, and take the infimum of total costs of chains joining two points. The <dyadic product estimate for group neighbourhoods> makes this a genuine compatible <metric>. The Hausdorff assumption separates distinct points; without it the same construction can give only a pseudometric.
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