Uniform Space - Examples

Examples

1. Every metric space (M, d) can be considered as a uniform space. Indeed, since a metric is a fortiori a pseudometric, the pseudometric definition furnishes M with a uniform structure. A fundamental system of entourages of this uniformity is provided by the sets
   This uniform structure on M generates the usual metric space topology on M. 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.
2. Using metrics, a simple example of distinct uniform structures with coinciding topologies can be constructed. For instance, let d₁(x,y) = | x − y | be the usual metric on R and let d₂(x,y) = | ex − ey |. Then both metrics induce the usual topology on R, yet the uniform structures are distinct, since { (x,y) : | x − y | < 1 } is an entourage in the uniform structure for d₁ but not for d₂. 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.
3. Every topological group G (in particular, every topological vector space) becomes a uniform space if we define a subset V of G × G to be an entourage if and only if it contains the set { (x, y) : x⋅y−1 in U } for some neighborhood U of the identity element of G. This uniform structure on G is called the right uniformity on G, because for every a in G, the right multiplication x → x⋅a is uniformly continuous with respect to this uniform structure. One may also define a left uniformity on G; the two need not coincide, but they both generate the given topology on G.
4. For every topological group G and its subgroup H the set of left cosets G/H is a uniform space with respect to the uniformity Φ defined as follows. The sets, where U runs over neighborhoods of the identity in G, form a fundamental system of entourages for the uniformity Φ. The corresponding induced topology on G/H is equal to the quotient topology defined by the natural map G → G/H.