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
- 2015: Barry Simon: Real Analysis: A Comprehensive Course in Analysis, Part 1: $2$ Topological Spaces: $2$ of Problems