Editing Topological group (section)

==Complete topological group{{anchor|Complete abelian topological groups}}==

{{See also|Complete uniform space}}

Information about convergence of nets and filters, such as definitions and properties, can be found in the article about [[filters in topology]].

===Canonical uniformity on a commutative topological group===

{{Main|Uniform space}}

This article will henceforth assume that any topological group that we consider is an additive commutative topological group with identity element <math>0.</math>

The '''diagonal''' of <math>X</math> is the set
<math display=block>\Delta_X := \{(x, x) : x \in X\}</math>
and for any <math>N \subseteq X</math> containing <math>0,</math> the '''canonical entourage''' or '''canonical vicinities around <math>N</math>''' is the set
<math display=block>\Delta_X(N) := \{(x, y) \in X \times X : x - y \in N\} = \bigcup_{y \in X} [(y + N) \times \{y\}] = \Delta_X + (N \times \{0\})</math>

For a topological group <math>(X, \tau),</math> the '''canonical uniformity'''{{sfn|Edwards|1995|p=61}} on <math>X</math> is the [[Uniform space|uniform structure]] induced by the set of all canonical entourages <math>\Delta(N)</math> as <math>N</math> ranges over all neighborhoods of <math>0</math> in <math>X.</math>

That is, it is the upward closure of the following prefilter on <math>X \times X,</math> 
<math display=block>\left\{\Delta(N) : N \text{ is a neighborhood of } 0 \text{ in } X\right\}</math>
where this prefilter forms what is known as a [[base of entourages]] of the canonical uniformity. 

For a commutative additive group <math>X,</math> a fundamental system of entourages <math>\mathcal{B}</math> is called a '''translation-invariant uniformity''' if for every <math>B \in \mathcal{B},</math> <math>(x, y) \in B</math> if and only if <math>(x + z, y + z) \in B</math> for all <math>x, y, z \in X.</math> A uniformity <math>\mathcal{B}</math> is called '''translation-invariant''' if it has a base of entourages that is translation-invariant.{{sfn|Schaefer|Wolff|1999|pp=12-19}}

<ul>
<li>The canonical uniformity on any commutative topological group is translation-invariant.</li>
<li>The same canonical uniformity would result by using a neighborhood basis of the origin rather the filter of all neighborhoods of the origin.</li>
<li>Every entourage <math>\Delta_X(N)</math> contains the diagonal <math>\Delta_X := \Delta_X(\{0\}) = \{(x, x) : x \in X\}</math> because <math>0 \in N.</math>
</li>
<li>If <math>N</math> is [[Symmetric set|symmetric]] (that is, <math>-N = N</math>) then <math>\Delta_X(N)</math> is symmetric (meaning that <math>\Delta_X(N)^{\operatorname{op}} = \Delta_X(N)</math>) and
<math>\Delta_X(N) \circ \Delta_X(N) = \{(x, z) : \text{ there exists } y \in X \text{ such that } x, z \in y + N\} = \bigcup_{y \in X} [(y + N) \times (y + N)] = \Delta_X + (N \times N).</math>
<li>The topology induced on <math>X</math> by the canonical uniformity is the same as the topology that <math>X</math> started with (that is, it is <math>\tau</math>).</li>
</ul>

===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}}

===Complete commutative topological group===

{{Main|Complete uniform space}}

Recall that for any <math>S \subseteq X,</math> a prefilter <math>\mathcal{C}</math> '''''on <math>S</math>''''' is necessarily a subset of <math>\wp(S)</math>; that is, <math>\mathcal{C} \subseteq \wp(S).</math>

A subset <math>S</math> of a topological group <math>X</math> is called a '''complete subset''' if it satisfies any of the following equivalent conditions: 
<ol>
<li>Every Cauchy prefilter <math>\mathcal{C} \subseteq \wp(S)</math> on <math>S</math> [[Filters in topology|converges]] to at least one point of <math>S.</math>
* If <math>X</math> is Hausdorff then every prefilter on <math>S</math> will converge to at most one point of <math>X.</math> But if <math>X</math> is not Hausdorff then a prefilter may converge to multiple points in <math>X.</math> The same is true for nets.</li>
<li>Every Cauchy net in <math>S</math> converges to at least one point of <math>S</math>;</li>
<li>Every Cauchy filter <math>\mathcal{C}</math> on <math>S</math> converges to at least one point of <math>S.</math></li>
<li><math>S</math> is a [[Complete uniform space|complete]] uniform space (under the point-set topology definition of "[[complete uniform space]]") when <math>S</math> is endowed with the uniformity induced on it by the canonical uniformity of <math>X</math>;</li>
</ol>

A subset <math>S</math> is called a '''sequentially complete subset''' if every Cauchy sequence in <math>S</math> (or equivalently, every elementary Cauchy filter/prefilter on <math>S</math>) converges to at least one point of <math>S.</math>

* Importantly, '''convergence outside of <math>S</math> is allowed''': If <math>X</math> is not Hausdorff and if every Cauchy prefilter on <math>S</math> converges to some point of <math>S,</math> then <math>S</math> will be complete even if some or all Cauchy prefilters on <math>S</math> ''also'' converge to points(s) in the complement <math>X \setminus S.</math> In short, there is no requirement that these Cauchy prefilters on <math>S</math> converge ''only'' to points in <math>S.</math> The same can be said of the convergence of Cauchy nets in <math>S.</math> 
** As a consequence, if a commutative topological group <math>X</math> is ''not'' [[Hausdorff space|Hausdorff]], then every subset of the closure of <math>\{0\},</math> say <math>S \subseteq \operatorname{cl} \{0\},</math> is complete (since it is clearly compact and every compact set is necessarily complete). So in particular, if <math>S \neq \varnothing</math> (for example, if <math>S</math> a is singleton set such as <math>S = \{0\}</math>) then <math>S</math> would be complete even though ''every'' Cauchy net in <math>S</math> (and every Cauchy prefilter on <math>S</math>), converges to ''every'' point in <math>\operatorname{cl} \{0\}</math> (include those points in <math>\operatorname{cl} \{0\}</math> that are not in <math>S</math>). 
** This example also shows that complete subsets (indeed, even compact subsets) of a non-Hausdorff space may fail to be closed (for example, if <math>\varnothing \neq S \subseteq \operatorname{cl} \{0\}</math> then <math>S</math> is closed if and only if <math>S = \operatorname{cl} \{0\}</math>).

A commutative topological group <math>X</math> is called a '''complete group''' if any of the following equivalent conditions hold:
<ol>
<li><math>X</math> is complete as a subset of itself.</li>
<li>Every Cauchy net in <math>X</math> [[Net (mathematics)|converges]] to at least one point of <math>X.</math></li>
<li>There exists a neighborhood of <math>0</math> in <math>X</math> that is also a complete subset of <math>X.</math>{{sfn|Narici|Beckenstein|2011|pp=48-51}}
* This implies that every locally compact commutative topological group is complete.</li>
<li>When endowed with its canonical uniformity, <math>X</math> becomes is a [[complete uniform space]].
* In the general theory of [[uniform space]]s, a uniform space is called a [[complete uniform space]] if each Cauchy [[Net (mathematics)|filter]] in <math>X</math> converges in <math>(X, \tau)</math> to some point of <math>X.</math></li>
</ol>

A topological group is called '''sequentially complete''' if it is a sequentially complete subset of itself.

'''Neighborhood basis''': Suppose <math>C</math> is a completion of a commutative topological group <math>X</math> with <math>X \subseteq C</math> and that <math>\mathcal{N}</math> is a [[neighborhood base]] of the origin in <math>X.</math> Then the family of sets 
<math display=block>\left\{ \operatorname{cl}_C N : N \in \mathcal{N} \right\}</math>
is a neighborhood basis at the origin in <math>C.</math>{{sfn|Narici|Beckenstein|2011|pp=47-66}}

'''{{visible anchor|Uniform continuity}}'''

Let <math>X</math> and <math>Y</math> be topological groups, <math>D \subseteq X,</math> and <math>f : D \to Y</math> be a map. Then <math>f : D \to Y</math> is '''uniformly continuous''' if for every neighborhood <math>U</math> of the origin in <math>X,</math> there exists a neighborhood <math>V</math> of the origin in <math>Y</math> such that for all <math>x, y \in D,</math> if <math>y - x \in U</math> then <math>f(y) - f(x) \in V.</math>