Uniform structure of a topological group

Summary

Every topological group carries natural uniform structures that induce its topology. The left uniformity declares $x$ and $y$ close when $x^{-1} y$ is close to the identity, and the right uniformity uses $y x^{-1}$ instead. These uniformities give meaning to Cauchy nets, uniform continuity and completeness without any metric. For abelian groups the two uniformities agree.