uniform structure of a topological group

Let G be a topological groupMathworldPlanetmath. There is a natural uniform structure on G which induces its topologyMathworldPlanetmath. We define a subset V of the Cartesian product G×G to be an entourage if and only if it contains a subset of the form


for some N neighborhoodMathworldPlanetmathPlanetmath of the identity elementMathworldPlanetmath. This is called the right uniformity of the topological group, with which multiplication becomes a uniformly continuous map. The left uniformity is defined in a fashion, but in general they don’t coincide, although they both induce the same topology on G.

Title uniform structure of a topological group
Canonical name UniformStructureOfATopologicalGroup
Date of creation 2013-03-22 12:47:21
Last modified on 2013-03-22 12:47:21
Owner mps (409)
Last modified by mps (409)
Numerical id 10
Author mps (409)
Entry type Derivation
Classification msc 54E15
Defines right uniformity
Defines left uniformity