Editing Complete metric space (section)

==Alternatives and generalizations==
{{Main|Uniform space#Completeness}}

Since [[Cauchy sequence]]s can also be defined in general [[topological group]]s, an alternative to relying on a metric structure for defining completeness and constructing the completion of a space is to use a group structure.  This is most often seen in the context of [[topological vector space]]s, but requires only the existence of a continuous "subtraction" operation.  In this setting, the distance between two points <math>x</math> and <math>y</math> is gauged not by a real number <math>\varepsilon</math> via the metric <math>d</math> in the comparison <math>d(x, y) < \varepsilon,</math> but by an [[open neighbourhood]] <math>N</math> of <math>0</math> via subtraction in the comparison <math>x - y \in N.</math>

A common generalisation of these definitions can be found in the context of a [[uniform space]], where an [[Uniform space#Entourage definition|entourage]] is a set of all pairs of points that are at no more than a particular "distance" from each other.

It is also possible to replace Cauchy ''sequences'' in the definition of completeness by Cauchy ''[[Net (mathematics)|nets]]'' or Cauchy [[Filter (set theory)#Filter|filters]].  If every Cauchy net (or equivalently every Cauchy filter) has a limit in <math>X,</math> then <math>X</math> is called complete.  One can furthermore construct a completion for an arbitrary uniform space similar to the completion of metric spaces. The most general situation in which Cauchy nets apply is [[Cauchy space]]s; these too have a notion of completeness and completion just like uniform spaces.