Editing Topological group (section)

===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>