An intersection inequality in groups

Question

Do you have an example of a group $G$ and decreasing sequences $(A_n),~(B_n)$ of its subsets such that $$\big(\bigcap_{i\in \Bbb N}A_i \big)\big( \bigcap_{j\in \Bbb N}B_j\big)\ne \bigcap_{i\in \Bbb N}\bigcap_{j\in \Bbb N}A_i B_j$$ ?

Answer 1

Let $G$ be the semi-direct product of $\mathbb{Z}$ acting on $\mathbb{Z}[\tfrac12]$ where the generator of the first acts as multiplication by $\tfrac12$ on the second. Then take $A_i$ from the first, decreasing to the trivial subgroup, and $B_j$ from the second decreasing to the trivial subgroup. Then $A_iB_j=G$ for all $i,j$, while the LHS is the trivial subgroup.