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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.

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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 9:55
    
@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
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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

1 Answer 1

up vote 6 down vote accepted

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 $1$.

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