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 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.
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 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.