Open subsets in topological groups I'm starting to study topological groups, and I noticed that every single theorem in topological groups I have to use the following statement:
Let $G$ be a topological group and U an open subset of G, if $g\in G$, then $gU$ is an open subset of G.
I can't prove it, please anyone can help me please.
Thanks

Multiplication is continuous, so the map 
$g^{-1}:G\to G, u\mapsto g^{-1}u$ is continuos. Thus the preimage of an open set $U\subset G$ is open, and the preimage equals $gU$.
Hope I got it right.
