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I am not sure if this is already posted, though, I hope I can get some help, and thank in advance. This question arises from the proof of the following.

Proposition: Let G be a topological group, of which H is a subgroup. Then, H is closed in G, if, and only if, there exists a neighborhood U of 1 in G, such that the intersection of U and H is closed in G.

During the proof, there is an assumption that confuses me, that is, it assumes a neighborhood V of 1, such that V=$V^{\iota}$, and that V*V is a subset of U. But, as far as I am concerned, there is no use of the assumption that $V^{\iota}$=V, and this is exactly my question.

Is it true that we always can find, for every neighborhood U of 1 in G, a neighborhood V of 1 in G such that V=$V^{\iota}$, and that V*V is a subset of U?

I know that the latter assumption results from the continuity of the multiplication, but I am very confused by the former. Thanks and regards here.

awllower
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    Choose $W$ containing $1$ such that $W\cdot W \subset U$. Then put $V = W \cap W^{-1}$. Are you by any chance reading Stroppel? Ugh, this notation... – t.b. Jun 29 '11 at 09:55
  • @Theo Buehler: Thanks a lot. Indeed, as you guess, I am reading the book on the locally compact groups by Stroppel; in view of your words, the notations seem very unusual. In any case, thank you very much. – awllower Jun 29 '11 at 10:27
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    Yes, indeed, you could call them "unusual", my interjection was intended to express that. Fortunately, tastes differ. I learned my basics on topological groups from the classic books by Weil (of whom you are a big admirer, as far as I seem to recall), Pontryagin and Hewitt-Ross (this one is also quite rough, notation-wise). – t.b. Jun 29 '11 at 10:32
  • @Theo Buehler: You, judged from the implications, must know how impressed, and surprised, I must have been, to find that there exists a book by Weil on this topic; this is just like a dream, which, fortunately enough for me, just comes true! Therefore, it is totally nature then for me to respectfully ask, whether or not, I have such a fortune, to know what the book is? In addition, if I was right, it is the basic number theory? I hope, with some evidence, I was wrong; for it does not appear to be a good introductory book, I might misunderstand something. Best regards here. – awllower Jun 30 '11 at 16:35
  • No, I meant L'intégration dans les groupes topologiques et ses applications, Actualités Scientifiques et Industrielles, 869, Paris: Hermann, 1940. I don't know if there exists a translation -- a quick Goolge search didn't reveal one. Yes, Basic number theory is about everything but basic in the sense of "elementary". L'integration is a bit easier to read, I think. Maybe you also want to look at Bourbaki's topologie générale, which contains many of the basic facts used and described by Stroppel, certainly (as Weil is one of the major contributors to the conception of that book)... – t.b. Jun 30 '11 at 17:26
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    ...most of the things relevant for number theory are covered there. Finally, I'd like to point out that L'intégration has a special status, for it was completed while Weil was in prison for deserting the country in order to avoid doing military service in the late 30ies. You should read Weil's autobiographical text Souvenirs d'apprentissage, where he describes the entire story. Highly recommended. – t.b. Jun 30 '11 at 17:30
  • I forgot to add: As far as I remember, you don't need much more than some basic facts from L'intégration to read Basic number theory, at least that's how Weil intended it to be according to the preface. – t.b. Jun 30 '11 at 19:14
  • Theo Buehler: Thank you for the time spent to explain; in fact, I tried, for quite a while, to read the Souvenirs d'apprentissage, and I actually have one in my disposal. However, after some efforts, I ended being exhausted by the repeated looking up in the dictionary; it requires too much in French reading for me; and I had decided to complete the reading this summer vacation, so, I will try, and thanks. Besides, I thought it was a paper, instead of a book; how ignorant I was... In fact, I read Stroppel in order to understand the basics of the Basic Number theory, but now I have other ... – awllower Jul 01 '11 at 09:49
  • options; therefore, I present here the most respect. From the point of view of me, it seems that vous is also a big fans of André Weil, doesn't it? To sum up, it is a great pleasure for me to be helped, thanks. Best regards here. – awllower Jul 01 '11 at 09:53
  • I hope you are aware that Souvenirs d'apprentissage exists in English translation. Here's an extensive review in the Notices of the AMS. In fact, the entire issue is dedicated to Weil's memory. Added: Yes, I spent a lot of time reading many of his works. This is due to the influence of my advisor who is a big fan of Weil as well and knew him personally from his time at Princeton. Best wishes back, – t.b. Jul 01 '11 at 09:54
  • Dear awllower, In case you haven't seen it already, I wanted to draw your attention to this thread that will most likely be of great interest to you. Best wishes, – t.b. Jul 12 '11 at 16:26
  • Dear Theo Buehler: Thank you fr that remind, and, of course, I just restarted the reading of Souvenirs d'appretissage once again, and gained still much pleasure therein, which might be the cause for me not to see the post. In addition, as I only gained some knowledge of the Haar integral from the book by Markus Stroppel, ans as Weil uses another notion, the Haar measure, to start from, I may want to know the relations between them. From the contexts, or the name itself, one can indeed derive some conjectures, but I still need a formal definition, and, thus, is still waiting for L'int.... – awllower Jul 13 '11 at 10:05
  • ...égration dans les groupes topologiques et ses applications. – awllower Jul 13 '11 at 10:06
  • I don't know what exactly you mean, but I suspect you are asking about the correspondence between positive linear functionals on the functions of compact support and Radon measures. This is called the Riesz-Markov theorem and can be found e.g. as Theorem 6.3.4 of Pedersen's Analysis now. Pedersen also wrote what I think is the best (and most efficient) exposition on existence and uniqueness of the Haar measure, which I uploaded to my homepage for your convenience. – t.b. Jul 13 '11 at 10:18
  • Dear Theo Buehler: Sorry for the absence recently, I was occupied by something. Although I not yet know what the Radon measures are, I thought that might be what I am trying to understand; moreover, the link in your post is indeed interesting, and I am ready to refresh my mind! Thank you very much. Also, best regards here. – awllower Jul 20 '11 at 11:34

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For the sake of having an answer: Choose a neighborhood $W$ of $1$ such that $W\cdot W \subset U$. This is possible by continuity of the multiplication, as you say. Then $V = W \cap W^{-1}$ is symmetric, i.e., $V^{-1} = V$ and $V\cdot V \subset W \cdot W \subset U$. As inversion is continuous and preserves the identity, $V$ is a neighborhood of $1$.

t.b.
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