When is a topological group Hausdorff (separated)? 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.
 A: You can probably find this result in a million places, one of which is N. Bourbaki, General Topology, Part 1, Chapter 3, Section 1.2, p. 223, Proposition 2.
A: 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: 


*

*{$0$} is closed.

*{$0$} is an intersection of the neighborhoods of zero. 

*$G$ is Hausdorff.

*$G$ is regular. 



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.

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 

A: Bourbaki, General Topology, III.2.5, prop 13. This is from an answer to this question.
