left cosets is a uniform space with respect to the uniformity defined as follows. The sets where runs over neighborhoods of the identity in form a fundamental system of entourages for the uniformity The corresponding induced topology on is equal to the quotient topology defined by the natural map The trivial topology belongs to a uniform space in which the whole cartesian product is the only entourage. History Before André Weil gave the first explicit definition of a uniform structure in 1937, uniform concepts, like completeness, were discussed using metric spaces. Nicolas Bourbaki provided the definition of uniform structure in terms of entourages in the book Topologie Générale and John Tukey gave the uniform cover definition. Weil also characterized uniform spaces in terms of a family of pseudometrics. See also References Nicolas Bourbaki, General Topology (Topologie Générale), (Ch. 1–4), (Ch. 5–10): Chapter II is a comprehensive reference of uniform structures, Chapter IX § 1 covers pseudometrics, and Chapter III § 3 covers uniform structures on topological groups Ryszard Engelking, General Topology. Revised and completed edition, Berlin 1989. John R. Isbell, Uniform Spaces I. M. James, Introduction to Uniform Spaces I. M. James, Topological and Uniform Spaces John Tukey, Convergence and Uniformity in Topology; André Weil, Sur les espaces à structure uniforme et sur la topologie générale, Act. Sci. Ind. 551, Paris, 1937