Birkhoff-Kakutani theorem

0.1 Birkhoff-Kakutani theorem

Theorem 0.1.

A topological groupMathworldPlanetmath (G,*,e) is metrizable if and only if G is HausdorffPlanetmathPlanetmath and the identity e of G has a countableMathworldPlanetmath neighborhoodMathworldPlanetmathPlanetmath basis. Here * is the group compositionMathworldPlanetmathPlanetmath law or operationMathworldPlanetmath. 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.


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