Let $G$ be an open subset of $\mathbb{R}$ . Then:
a) Is the set $H=\{xy|x,y\in G\ \text{and}\ 0\notin G\}$ open in $\mathbb{R}$?
b) Is the set $G=\mathbb{R}$ if $0\in G$ and $\forall x,y\in G, x+y\in G$?
I think the answer to both the problems is yes. But, the question is, should we use the group theoretic properties or topological properties to prove the statements? And how should we exactly proceed. Any hints? Thanks beforehand.