Basis Element of Furstenberg Topology is Clopen

Theorem
Let $\tau$ be the Furstenberg topology on the set of integers $\Z$.

Let $a, b \in \Z$ such that $a \ne 0$.

Then $a \Z + b$ is clopen in $\struct {\Z, \tau}$.

Proof
$a \Z + b \in \tau$ by.

It remains to show:
 * $\Z \setminus \paren {a \Z + b} \in \tau$

As $a \Z = \paren {-a} \Z$, we may assume $a > 0$.

If $a = 1$, then $\Z \setminus \Z = \O \in \tau$.

Thus we assume that $a \ge 2$.

Then:

That is:
 * $\ds \Z \setminus \paren {a \Z + b} = \bigcup _{k \mathop \in \set {1, \ldots, a-1} } a \Z + k$

where $a \Z + k \in \tau$ for all $k \in \set {1, \ldots, a-1}$.

Therefore $\Z \setminus \paren {a \Z + b} \in \tau$.