Definition:Furstenberg Topology

Definition
Let $\Z$ be the set of integers.

Let:
 * $\BB := \set {a \Z + b : a,b \in \Z, a \ne 0}$

where:
 * $a \Z + b := \set {a k + b : k \in \Z}$

Let:
 * $\ds \tau := \set { \bigcup \AA : \AA \subseteq \BB }$

Then $\tau$ is called Furstenberg topology on $\Z$.

Also known as
This topology is also called the arithmetic progression topology, as $a \Z + b$ are doubly infinite arithmetic progressions.

Also see

 * Furstenberg Topology is Topology: Especially, it is shown that $\BB$ is a synthetic basis on $\Z$.