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 Definition of Furstenberg Topology.

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:

\(\ds x\)

\(\in\)

\(\ds \Z \setminus \paren {a \Z + b}\)

\(\ds \leadstoandfrom \ \ \)

\(\ds x - b\)

\(\not \in\)

\(\ds a \Z\)

\(\ds \leadstoandfrom \ \ \)

\(\ds x - b\)

\(\not \equiv\)

\(\ds 0\)

\(\ds \pmod a\)

\(\ds \leadstoandfrom \ \ \)

\(\ds \forall k \in \set {1, \ldots , a-1}: \, \)

\(\ds x - b\)

\(\equiv\)

\(\ds k\)

\(\ds \pmod a\)

\(\ds \leadstoandfrom \ \ \)

\(\ds \forall k \in \set {1, \ldots , a-1}: \, \)

\(\ds x - b\)

\(\in\)

\(\ds a \Z + k\)

\(\ds \leadstoandfrom \ \ \)

\(\ds x\)

\(\in\)

\(\ds \bigcup _{k \mathop \in \set {1, \ldots , a-1} } a \Z + k\)

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

$\blacksquare$