Definition:Furstenberg Topology

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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:

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


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

  • Results about the Furstenberg topology can be found here.


Source of Name

This entry was named for Hillel Furstenberg.


Sources