Exercise 1, § 2

Let $K$ be a commutative field, $A$ the polynomial ring $K[X, Y]$ in two indeterminates and $a$, the principal ideal $(XY^n)$ in $A$; the sequence $(a_n)_{n > 0}$, forms with $a_n = A$ an exhaustive and separated filtration on $A$. Let $b$ be the principal ideal $(X)$ of $A$; show that the topology on $b$ induced by that on $A$ is strictly coarser than the topology defined by the filtration on $b$ associated with that on $A$ (and a fortiori this latter filtration is distinct from the filtration induced by that on $A$).