Question

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.

Answer 1

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

Answer 2

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.

Answer 3

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