The neighborhood of e in a topological group

Question

suppse U is an open neighborhood of e in a topological group G and how to prove there exists an open neighborhood V s. t. $V=V^{-1}$ and $V^{2}$ is a subset of U? Another question is.how to prove the union of all powers of U is a closed subset of G?

Answer 1

I'll give you a hint:- the multiplication map $\mu\colon G\times G\to G$ is continuous and $\mu^{-1}(U)\ni (e,e)$, so you can find open subsets $W_1,W_2$ of $G$ both containing $e$ such that $W_1\times W_2\subseteq\mu^{-1}(U)$. Now, what do you know about $W:=W_1\cap W_2$? And what about $V:=W\cap W^{-1}$?