open subset of $G\times G$

Question

If $O$ be an open symmetric subset of topological group $G$ such that $e\in O$, then is $V_O=\{(a,b)\in G\times G: a^{-1}b\in O\}$ open in $G\times G$?

Answer 1

Yes. $f\colon G\times G\to G$, $(x,y)\mapsto x^{-1}y$ is continuous, hence $V_O=f^{-1}(O)$ is open. (We do not need $e\in O$ or symmetry of $O$ for this)

Answer 2

Let $m$ denote the multiplication in $G$, $i$ the inversion. Notice that $f = m(i(\cdot),\cdot)$ is continuous in $G\times G$. Then $V_0 = f^{-1}(O)$, and thus it is open.