suppose that i have a polish group $G$ and two subsets $A$ and $B$ of $G$ such that: $A$ is open in $G$ and $B$ is closed in $G,$ from this can i conclude that $AB$ is a Borel subset of $G$? if not, consider the following.

Since $B$ is closed, $B$ have a Borel transversal $T,$ that is  a Borel set $T$ such that $T\cap gB$ is a unitary set, for all $g\in G,$ then can i conclude that the set $AB\cap T$ is Borel?.


Any reference will help.
Thanks