Editing Topological group (section)

===Cauchy prefilters and nets===

{{Main|Filters in topology|Net (mathematics)}}

The general theory of [[uniform space]]s has its own definition of a "Cauchy prefilter" and "Cauchy net." For the canonical uniformity on <math>X,</math> these reduces down to the definition described below.

Suppose <math>x_{\bull} = \left(x_i\right)_{i \in I}</math> is a net in <math>X</math> and <math>y_{\bull} = \left(y_j\right)_{j \in J}</math> is a net in <math>Y.</math> Make <math>I \times J</math> into a directed set by declaring <math>(i, j) \leq \left(i_2, j_2\right)</math> if and only if <math>i \leq i_2 \text{ and } j \leq j_2.</math> Then{{sfn|Narici|Beckenstein|2011|pp=47-66}} <math>x_{\bull} \times y_{\bull}: = \left(x_i, y_j\right)_{(i, j) \in I \times J}</math> denotes the '''product net'''. If <math>X = Y</math> then the image of this net under the addition map <math>X \times X \to X</math> denotes the '''sum''' of these two nets:
<math display=block>x_{\bull} + y_{\bull}: = \left(x_i + y_j\right)_{(i, j) \in I \times J}</math>
and similarly their '''difference''' is defined to be the image of the product net under the subtraction map:
<math display=block>x_{\bull} - y_{\bull}: = \left(x_i - y_j\right)_{(i, j) \in I \times J}.</math>

A [[Net (mathematics)|net]] <math>x_{\bull} = \left(x_i\right)_{i \in I}</math> in an additive topological group <math>X</math> is called a '''Cauchy net''' if{{sfn|Narici|Beckenstein|2011|p=48}}
<math display=block>\left(x_i - x_j\right)_{(i, j) \in I \times I} \to 0 \text{ in } X</math>
or equivalently, if for every neighborhood <math>N</math> of <math>0</math> in <math>X,</math> there exists some <math>i_0 \in I</math> such that 
<math>x_i - x_j \in N</math> for all indices <math>i, j \geq i_0.</math> 

A '''[[Cauchy sequence]]''' is a Cauchy net that is a sequence.

If <math>B</math> is a subset of an additive group <math>X</math> and <math>N</math> is a set containing <math>0,</math> then<math>B</math> is said to be an '''<math>N</math>-small set''' or '''small of order <math>N</math>''' if <math>B - B \subseteq N.</math>{{sfn|Narici|Beckenstein|2011|pp=48-51}}

A prefilter <math>\mathcal{B}</math> on an additive topological group <math>X</math> called a '''Cauchy prefilter''' if it satisfies any of the following equivalent conditions: 
<ol>
<li><math>\mathcal{B} - \mathcal{B} \to 0</math> in <math>X,</math> where <math>\mathcal{B} - \mathcal{B} := \{B - C : B, C \in \mathcal{B}\}</math> is a prefilter.</li>
<li><math>\{B - B : B \in \mathcal{B}\} \to 0</math> in <math>X,</math> where <math>\{B - B : B \in \mathcal{B}\}</math> is a prefilter equivalent to <math>\mathcal{B} - \mathcal{B}.</math></li>
<li>For every neighborhood <math>N</math> of <math>0</math> in <math>X,</math> <math>\mathcal{B}</math> contains some <math>N</math>-small set (that is, there exists some <math>B \in \mathcal{B}</math> such that <math>B - B \subseteq N</math>).{{sfn|Narici|Beckenstein|2011|pp=48–51}}</li>
</ol>
and if <math>X</math> is commutative then also:
<ol start=4>
<li>For every neighborhood <math>N</math> of <math>0</math> in <math>X,</math> there exists some <math>B \in \mathcal{B}</math> and some <math>x \in X</math> such that <math>B \subseteq x + N.</math>{{sfn|Narici|Beckenstein|2011|pp=48-51}}</li>
</ol>
* It suffices to check any of the above condition for any given [[neighborhood basis]] of <math>0</math> in <math>X.</math>

Suppose <math>\mathcal{B}</math> is a prefilter on a commutative topological group <math>X</math> and <math>x \in X.</math> Then <math>\mathcal{B} \to x</math> in <math>X</math> if and only if <math>x \in \operatorname{cl} \mathcal{B}</math> and <math>\mathcal{B}</math> is Cauchy.{{sfn|Narici|Beckenstein|2011|pp=47-66}}