Page "Topological group" Paragraph 19

Every topological group can be viewed as a uniform space in two ways ; the left uniformity turns all left multiplications into uniformly continuous maps while the right uniformity turns all right multiplications into uniformly continuous maps.

If G is not abelian, then these two need not coincide.

The uniform structures allow one to talk about notions such as completeness, uniform continuity and uniform convergence on topological groups.