Editing Uniform space (section)

===Pseudometrics definition===

Uniform spaces may be defined alternatively and equivalently using systems of [[Pseudometric space|pseudometrics]], an approach that is particularly useful in [[functional analysis]] (with pseudometrics provided by [[seminorm]]s). More precisely, let <math>f : X \times X \to \R</math> be a pseudometric on a set <math>X.</math> The inverse images <math>U_a = f^{-1}([0, a])</math> for <math>a > 0</math> can be shown to form a fundamental system of entourages of a uniformity. The uniformity generated by the <math>U_a</math> is the uniformity defined by the single pseudometric <math>f.</math> Certain authors call spaces the topology of which is defined in terms of pseudometrics ''gauge spaces''.

For a ''family'' <math>\left(f_i\right)</math> of pseudometrics on <math>X,</math> the uniform structure defined by the family is the ''least upper bound'' of the uniform structures defined by the individual pseudometrics <math>f_i.</math> A fundamental system of entourages of this uniformity is provided by the set of ''finite'' intersections of entourages of the uniformities defined by the individual pseudometrics <math>f_i.</math> If the family of pseudometrics is ''finite'', it can be seen that the same uniform structure is  defined by a ''single'' pseudometric, namely the [[upper envelope]] <math>\sup_{} f_i</math> of the family.

Less trivially, it can be shown that a uniform structure that admits a [[countable]] fundamental system of entourages (hence in particular a uniformity defined by a countable family of pseudometrics) can be defined by a single pseudometric. A consequence is that ''any'' uniform structure can be defined as above by a (possibly uncountable) family of pseudometrics (see Bourbaki: General Topology Chapter IX §1 no. 4).