uniform structure of a topological group

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

$$V_N=\{(x,y)\in G\times G:x{y}^{-1}\in N\}$$

for some $N$ neighborhood of the identity element. 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