Exercise 13, § 2

Let $K$ be a field, $A = K[[X]]$ the ring of formal power series and $m$ its maximal ideal, so that $A$ is Hausdorff and complete with the $m$-adic topology (no. 6, Corollary to Proposition 6). On the additive group $A$ consider the filtration $(E_n)$ such that $E_0 = A$ and $E_n$ is the intersection of $m^n$ and the ring $K[X]$; this filtration is exhaustive and separated, the topology $\mathscr{T}$ which it defines on $A$ is compatible with the additive group structure on $A$ and is finer than the $m$-adic topology but $A$ is not a complete group with the topology $\mathscr{T}$ (consider the sequence of polynomials $(1 - X^n)/(1 - X)$).