the map can be defined by mapping to The map thus defined is in general not injective; in fact, the graph of the equivalence relation is the intersection of all entourages of and thus is injective precisely when is Hausdorff. The uniform structure on is defined as follows: for each (that is, such that implies ), let be the set of all pairs of minimal Cauchy filters which have in common at least one -small set. The sets can be shown to form a fundamental system of entourages; is equipped with the uniform structure thus defined. The set is then a dense subset of If is Hausdorff, then is an isomorphism onto and thus can be identified with a dense subset of its completion. Moreover, is always Hausdorff; it is called the If denotes the equivalence relation then the quotient space is homeomorphic to Examples Every metric space can be considered as a uniform space. Indeed, since a metric is a fortiori a pseudometric, the pseudometric definition furnishes with a uniform structure. A fundamental system of entourages of this uniformity is provided by the setsThis uniform structure on generates the usual metric space topology on However, different metric spaces can have the same uniform structure (trivial example is provided by a constant multiple of a metric). This uniform structure produces also equivalent definitions of uniform continuity and completeness for metric spaces. Using metrics, a simple example of distinct uniform structures with coinciding topologies can be constructed. For instance, let be the usual metric on and let Then both metrics induce the usual topology on yet the uniform structures are distinct, since is an entourage in the uniform structure for but not for Informally, this example can be seen as taking the usual uniformity and distorting it through the action of a continuous yet non-uniformly continuous function. Every topological group (in particular, every topological vector space) becomes a uniform space if we define a subset to be an entourage if and only if it contains the set for some neighborhood of the identity element of This uniform structure on is called the right uniformity on because for every the right multiplication is uniformly continuous with respect to this uniform structure. One may also define a left uniformity on the two need not coincide, but they both generate the given topology on For every topological group and its subgroup the set of