Editing Uniform space (section)

==Examples==

# Every [[metric space]] <math>(M, d)</math> can be considered as a uniform space. Indeed, since a metric is ''a fortiori'' a pseudometric, the [[Uniform space#Pseudometrics definition|pseudometric definition]] furnishes <math>M</math> with a uniform structure. A fundamental system of entourages of this uniformity is provided by the sets<blockquote><math> \qquad U_a \triangleq d^{-1}([0,a]) = \{(m, n) \in M \times M : d(m,n) \leq a\}.</math></blockquote>This uniform structure on <math>M</math> generates the usual metric space topology on <math>M.</math> 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 [[complete metric space|completeness for metric spaces]].
# Using metrics, a simple example of distinct uniform structures with coinciding topologies can be constructed.  For instance, let <math>d_1(x, y) = |x - y|</math> be the usual metric on <math>\R</math> and let <math>d_2(x, y) = \left|e^x - e^y\right|.</math>  Then both metrics induce the usual topology on <math>\R,</math> yet the uniform structures are distinct, since <math>\{(x, y) : |x - y| < 1\}</math> is an entourage in the uniform structure for <math>d_1(x, y)</math> but not for <math>d_2(x, y).</math>  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]] <math>G</math> (in particular, every [[topological vector space]]) becomes a uniform space if we define a subset <math>V \subseteq G \times G</math> to be an entourage if and only if it contains the set <math>\{(x, y) : x \cdot y^{-1} \in U\}</math> for some [[neighbourhood (topology)|neighborhood]] <math>U</math> of the [[identity element]] of <math>G.</math> This uniform structure on <math>G</math> is called the ''right uniformity'' on <math>G,</math> because for every <math>a \in G,</math> the right multiplication <math>x \to x \cdot a</math> is [[Uniform space#Uniform continuity|uniformly continuous]] with respect to this uniform structure. One may also define a left uniformity on <math>G;</math> the two need not coincide, but they both generate the given topology on <math>G.</math>
# For every topological group <math>G</math> and its subgroup <math>H \subseteq G</math> the set of left [[coset]]s <math>G / H</math> is a uniform space with respect to the uniformity <math>\Phi</math> defined as follows. The sets <math>\tilde{U} = \{(s,t) \in G/H \times G/H : \ \ t \in U \cdot s\},</math> where <math>U</math> runs over neighborhoods of the identity in <math>G,</math> form a fundamental system of entourages for the uniformity <math>\Phi.</math> The corresponding induced topology on <math>G / H</math> is equal to the [[quotient topology]] defined by the natural map <math>g \to G / H.</math>
# The trivial topology belongs to a uniform space in which the whole cartesian product <math>X \times X</math> is the only [[entourage (topology)|entourage]].