Birkhoff-Kakutani theorem
0.1 Birkhoff-Kakutani theorem
Theorem 0.1.
A topological group^{} $\mathrm{(}G\mathrm{,}\mathrm{*}\mathrm{,}e\mathrm{)}$ is metrizable if and only if $G$ is Hausdorff^{} and the identity $e$ of $G$ has a countable^{} neighborhood^{} basis. Here $\mathrm{*}$ is the group composition^{} law or operation^{}. 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.
References
- 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.
Title | Birkhoff-Kakutani theorem |
---|---|
Canonical name | BirkhoffKakutaniTheorem |
Date of creation | 2013-03-22 18:24:34 |
Last modified on | 2013-03-22 18:24:34 |
Owner | bci1 (20947) |
Last modified by | bci1 (20947) |
Numerical id | 17 |
Author | bci1 (20947) |
Entry type | Theorem |
Classification | msc 22A22 |
Classification | msc 22A10 |
Classification | msc 22A05 |
Related topic | TopologicalGroup2 |
Related topic | T2Space |
Related topic | HomotopyDoubleGroupoidOfAHausdorffSpace |