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Birkhoff-Kakutani theorem (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.



"Birkhoff-Kakutani theorem" is owned by bci1.

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Cross-references: metric, identity, topological group

This is version 1 of Birkhoff-Kakutani theorem, born on 2009-02-04.
Object id is 485, canonical name is BirkhoffKakutaniTheorem.
Accessed 357 times total.

Classification:
Physics Classification: 00. (GENERAL)
 02. (Mathematical methods in physics)
Pending Errata and Addenda
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