Let $\mathcal T$ be a topology on $\mathbb Z:$ $\mathcal T\subset\mathcal P(\mathbb Z)$ with $\emptyset$ and all the unions of $S(a,b)=\{an+b|n\in \mathbb Z\}$ for $a\neq 0$ prove that $\mathcal T$ is topology on the integers.
I know that I should show:
$\ \ \ $1) $\emptyset$ are in $\mathcal T$.
$\ \ \ $2) Any union of elements of $T$ belongs to $\mathcal T$.
$\ \ \ $3) Any finite intersection of elements of $\mathcal T$ belongs to $\mathcal T$
Have no idea how to start