Profinite topology

The profinite topology on a group takes the cosets of finite-index normal subgroups as a neighborhood basis. A profinite group carries the inverse-limit topology from its finite quotients.

The profinite topology on $\mathbb{Z}$ has basis the residue classes $a+k\mathbb{Z}$, $k\ge 1$. These classes are both open and closed, since their complements are finite unions of other residue classes. Set $F_n=(4^n-1)/3+4^n\mathbb{Z}$ for $n\ge 1$. Then $F_{n+1}\subseteq F_n$ and every $F_n$ is nonempty. An integer $x$ in their intersection would make $3x+1$ divisible by every $4^n$, forcing $3x+1=0$, impossible. Thus the open complements $\mathbb{Z}\setminus F_n$ form an increasing cover with no finite subcover. The topology is Hausdorff because distinct integers are distinct modulo a sufficiently large integer, but is not compact.

An inverse limit carries the coarsest topology making every coordinate projection map continuous. It is the subspace topology inherited from the product of the component spaces.