property of a topological group??

Hello;

Show that if is a first countable topological group, then there is a sequence of symmetric neighborhoods of the neutral element of such that

(1){ } is a local base at in G

(2) , for every

I understand number 1 is due to first countability of , but how can we get (2). Please guide me. Every comment or guidance is highly appreciated

Recall that a neighborhood of the identity element is said to be symmetric if it equal to its inverse.

Re: property of a topological group??

Did you try to use the continuity of the map , ?

Re: property of a topological group??

Ok, thanks a lot my instructor. Now, I understand the concept.

Let be any neighborhood of the identical element in . By continuity of there is a neighborhood of such that . By continuity of there is a neighborhood of such that . That's mean any neighborhood of contains a symmetric neighborhood. Now is a symmetric neighborhood of and . Because of that, we can construct such a sequence of symmetric neighborhood of .

Thank you very much for the very helpful guidance.(Clapping)