0.1 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,
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.
