Does someone knows a good reference for the following result?


"A topological group is Hausdorff if and only if the identity is closed."


I have seen proofs in lecture notes of courses on the web, but I would like a reference in a book or an article, in order to refer to it.



When is a topological group Hausdorff (separated)?
You could find a routin proof in the book "Topological Ring" written by Seth Warner. In this book at page 21 in Theorem 3.4 you could see the following Proposition:


Theorem: Let $G$ be a topological group. The following statements are equivalent: 


1. {$0$} is closed.
2. {$0$} is an intersection of the neighborhoods of zero.
3. $G$ is Hausdorff.
4. $G$ is regular.




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You could also find the improvement of it in the book "**Topology for analysis**" Written by **Albert Wilansky**. In section 12 at page 243 You could see the following theorem:


THEQREM: Every topological group is completely regular. The following conditions on 
a topological group $G$ are equivalent: 


* $G$ is a $T\_0$ space.
* $G$ is a Tychonoff space.-
* $\cap${$U:U$ is a is a neighborhood of $e$}={$e$}


The Proof of Complete regularity Has more details. I think The proof is in the level of Urysohn Lemma.




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But if you are interested in the general case, I Suggest You look at the section "uniformity" which were discussed in the following books:


* Topology for Analysis: Chapter 11
* General topology: Stephen Willard: chapter 9


