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PlanetPhysics/Birkhoff Kakutani Theorem

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Birkhoff-Kakutani theorem

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\begin{theorem}

A topological group is metrizable if and only if is Hausdorff and the identity of has a countable neighborhood basis. Furthermore, if G is metrizable , then admits a compatible metric which is left-invariant, that is, a right-invariant metric also exists under these conditions. \end{theorem}

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References

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  1. Howard Becker, Alexander S. Kechris. 1996. The Descriptive Set Theory of Polish Group Actions. (London Mathematical Society Lecture Note Series) , Cambridge University Press: Cambridge, UK, p.14.