Editing Uniform space (section)

==Completeness==

Generalizing the notion of [[complete metric space]], one can also define completeness for uniform spaces. Instead of working with [[Cauchy sequence]]s, one works with [[Cauchy filter]]s (or [[Cauchy net]]s).

A '''{{em|{{visible anchor|Cauchy filter}}}}''' (respectively, a '''{{em|{{visible anchor|Cauchy prefilter}}}}''') <math>F</math> on a uniform space <math>X</math> is a [[Filter (set theory)|filter]] (respectively, a [[prefilter]]) <math>F</math> such that for every entourage <math>U,</math> there exists <math>A \in F</math> with <math>A \times A \subseteq U.</math> In other words, a filter is Cauchy if it contains "arbitrarily small" sets. It follows from the definitions that each filter that converges (with respect to the topology defined by the uniform structure) is a Cauchy filter.
A '''{{em|{{visible anchor|minimal Cauchy filter}}}}''' is a Cauchy filter that does not contain any smaller (that is, coarser) Cauchy filter (other than itself). It can be shown that every Cauchy filter contains a unique {{em|minimal Cauchy filter}}. The neighbourhood filter of each point (the filter consisting of all neighbourhoods of the point) is a minimal Cauchy filter.

Conversely, a uniform space is called '''{{em|{{visible anchor|text=complete|Complete uniform space|Complete space}}}}''' if every Cauchy filter converges. Any compact Hausdorff space is a complete uniform space with respect to the unique uniformity compatible with the topology.

Complete uniform spaces enjoy the following important property: if <math>f : A \to Y</math> is a ''uniformly continuous'' function from a [[Dense set|''dense'' subset]] <math>A</math> of a uniform space <math>X</math> into a ''complete'' uniform space <math>Y,</math> then <math>f</math> can be extended (uniquely) into a uniformly continuous function on all of <math>X.</math>

A topological space that can be made into a complete uniform space, whose uniformity induces the original topology, is called a [[completely uniformizable space]].

A {{visible anchor|completion|completion of a uniform space|text='''{{em|completion}}''' of a uniform space}} <math>X</math> is a pair <math>(i, C)</math> consisting of a complete uniform space <math>C</math> and a [[#uniform embedding|uniform embedding]] <math>i : X \to C</math> whose image <math>i(X)</math> is a [[Dense set|dense subset]] of <math>C.</math> 

===Hausdorff completion of a uniform space===

As with metric spaces, every uniform space <math>X</math> has a {{visible anchor|Hausdorff completion|Hausdorff completion of a uniform space|text='''{{em|Hausdorff completion}}'''}}: that is, there exists a complete Hausdorff uniform space <math>Y</math> and a uniformly continuous map <math>i : X \to Y</math> (if <math>X</math> is a Hausdorff uniform space then <math>i</math> is a [[topological embedding]]) with the following property:

: for any uniformly continuous mapping <math>f</math> of <math>X</math> into a complete Hausdorff uniform space <math>Z,</math> there is a unique uniformly continuous map <math>g : Y \to Z</math> such that <math>f = g i.</math>

The Hausdorff completion <math>Y</math> is unique up to isomorphism. As a set, <math>Y</math> can be taken to consist of the {{em|minimal}} Cauchy filters on <math>X.</math> As the neighbourhood filter <math>\mathbf{B}(x)</math> of each point <math>x</math> in <math>X</math> is a minimal Cauchy filter, the map <math>i</math> can be defined by mapping <math>x</math> to <math>\mathbf{B}(x).</math> The map <math>i</math> thus defined is in general not injective; in fact, the graph of the equivalence relation <math>i(x) = i(x')</math> is the intersection of all entourages of <math>X,</math> and thus <math>i</math> is injective precisely when <math>X</math> is Hausdorff.

The uniform structure on <math>Y</math> is defined as follows: for each {{visible anchor|symmetric entourage|text=''symmetric'' entourage}} <math>V</math> (that is, such that <math>(x, y) \in V</math> implies <math>(y, x) \in V</math>), let <math>C(V)</math> be the set of all pairs <math>(F, G)</math> of minimal Cauchy filters ''which have in common at least one <math>V</math>-small set''. The sets <math>C(V)</math> can be shown to form a fundamental system of entourages; <math>Y</math> is equipped with the uniform structure thus defined.

The set <math>i(X)</math> is then a dense subset of <math>Y.</math>  If <math>X</math> is Hausdorff, then <math>i</math>  is an isomorphism onto <math>i(X),</math> and thus <math>X</math> can be identified with a dense subset of its completion. Moreover, <math>i(X)</math> is always Hausdorff; it is called the {{visible anchor|associated Hausdorff uniform space|text='''Hausdorff uniform space associated with''' <math>X.</math>}}  If <math>R</math> denotes the equivalence relation <math>i(x) = i(x'),</math> then the quotient space <math>X / R</math> is homeomorphic to <math>i(X).</math>