# 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

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

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 UniformStructureOfATopologicalGroup 2013-03-22 12:47:21 2013-03-22 12:47:21 mps (409) mps (409) 10 mps (409) Derivation msc 54E15 right uniformity left uniformity