Editing Topological group (section)

=== Separation properties ===

If {{mvar|U}} is an open subset of a commutative topological group {{mvar|G}} and {{mvar|U}} contains a compact set {{mvar|K}}, then there exists a neighborhood {{mvar|N}} of the identity element such that {{math|''KN'' ⊆ ''U''}}.{{sfn|Narici|Beckenstein|2011|pp=19-45}}

As a uniform space, every commutative topological group is [[completely regular space|completely regular]]. 
Consequently, for a multiplicative topological group {{mvar|G}} with identity element 1, the following are equivalent:{{sfn|Narici|Beckenstein|2011|pp=19-45}} 
<ol>
<li>{{mvar|G}} is a T<sub>0</sub>-space ([[Kolmogorov space|Kolmogorov]]);</li>
<li>{{mvar|G}} is a T<sub>2</sub>-space ([[Hausdorff space|Hausdorff]]);</li>
<li>{{mvar|G}} is a T<sub>3{{frac|1|2}}</sub> ([[Tychonoff space|Tychonoff]]);</li>
<li>{{math|{ 1 } }} is closed in {{mvar|G}};</li>
<li>{{math|{ 1 } :{{=}} {{underset|N ∈ 𝒩|{{big|∩}}}} ''N''}}, where {{math|𝒩}} is a neighborhood basis of the identity element in {{mvar|G}};</li>
<lI>for any <math>x \in G</math> such that <math>x \neq 1,</math> there exists a neighborhood {{mvar|U}} in {{mvar|G}} of the identity element such that <math>x \not\in U.</math></li>
</ol>

A subgroup of a commutative topological group is discrete if and only if it has an [[isolated point]].{{sfn|Narici|Beckenstein|2011|pp=19-45}}

If {{mvar|G}} is not Hausdorff, then one can obtain a Hausdorff group by passing to the quotient group {{math|''G''/''K''}}, where {{mvar|K}} is the [[closure (topology)|closure]] of the identity.{{sfn|Bourbaki|1998|loc=section III.2.7}} 
This is equivalent to taking the [[Kolmogorov space#The Kolmogorov quotient|Kolmogorov quotient]] of {{mvar|G}}.