It's true that the space $\Bbb Z$ of integers cannot have a compact topology. But as a topological space with the discrete topology it does have many...
Let $R_n$ be the number consisting of $n$ $1$s. If $n>m$ then $R_n - R_m$ consists of $n-m$ $1$s followed by $m$ $0$s. So $R_n - R_m = 10^mR_{n-m}$....