When is a topological group Hausdorff (separated)? - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-20T05:45:27Z http://mathoverflow.net/feeds/question/99478 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/99478/when-is-a-topological-group-hausdorff-separated When is a topological group Hausdorff (separated)? Jérémy Blanc 2012-06-13T17:01:46Z 2012-06-13T20:02:59Z <p>Does someone knows a good reference for the following result?</p> <p>"A topological group is Hausdorff if and only if the identity is closed."</p> <p>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.</p> http://mathoverflow.net/questions/99478/when-is-a-topological-group-hausdorff-separated/99480#99480 Answer by Greg Marks for When is a topological group Hausdorff (separated)? Greg Marks 2012-06-13T17:31:03Z 2012-06-13T17:31:03Z <p>You can probably find this result in a million places, one of which is N. Bourbaki, <i>General Topology</i>, Part 1, Chapter 3, Section 1.2, p. 223, Proposition 2.</p> http://mathoverflow.net/questions/99478/when-is-a-topological-group-hausdorff-separated/99481#99481 Answer by Igor Rivin for When is a topological group Hausdorff (separated)? Igor Rivin 2012-06-13T17:35:23Z 2012-06-13T17:35:23Z <p>Bourbaki, General Topology, III.2.5, prop 13. This is from an answer to <a href="http://mathoverflow.net/questions/9022/quotient-of-a-hausdorff-topological-group-by-a-closed-subgroup" rel="nofollow">this question</a>.</p> http://mathoverflow.net/questions/99478/when-is-a-topological-group-hausdorff-separated/99494#99494 Answer by AliReza Olfati for When is a topological group Hausdorff (separated)? AliReza Olfati 2012-06-13T20:02:59Z 2012-06-13T20:02:59Z <p>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:</p> <p>Theorem: Let \$G\$ be a topological group. The following statements are equivalent: </p> <ol> <li>{\$0\$} is closed.</li> <li>{\$0\$} is an intersection of the neighborhoods of zero. </li> <li>\$G\$ is Hausdorff.</li> <li>\$G\$ is regular. </li> </ol> <hr> <p>You could also find the improvement of it in the book "<strong>Topology for analysis</strong>" Written by <strong>Albert Wilansky</strong>. In section 12 at page 243 You could see the following theorem:</p> <p>THEQREM: Every topological group is completely regular. The following conditions on a topological group \$G\$ are equivalent: </p> <ul> <li>\$G\$ is a \$T_0\$ space.</li> <li>\$G\$ is a Tychonoff space.-</li> <li>\$\cap\${\$U:U\$ is a is a neighborhood of \$e\$}={\$e\$}</li> </ul> <p>The Proof of Complete regularity Has more details. I think The proof is in the level of Urysohn Lemma.</p> <hr> <p>But if you are interested in the general case, I Suggest You look at the section "uniformity" which were discussed in the following books:</p> <ul> <li>Topology for Analysis: Chapter 11</li> <li>General topology: Stephen Willard: chapter 9 </li> </ul>