Birkhoff-Kakutani theorem

Birkhoff-Kakutani theorem

Theorem 0.1  

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,

$\displaystyle d(gx, gy) = d(x,y);$

a right-invariant metric $ r$ also exists under these conditions.

Bibliography

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.



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