Definition 7.3 (Profinite topology). Assume $A_n$ is finite. The profinite topology on $A:=\underset{\stackrel{[}{n}]\leftarrow }{\lim }{A}_n$ is the weakest topology on $A$ such that ${𝜃}_n:A\to A_n$ is continuous for all $n$, where $A_n$ is equipped with the discrete topology.