Furstenberg Topology is Topology
Theorem
Let $\struct {\Z, \tau}$ be the topological space formed by the Furstenberg topology on the set of integers $\Z$.
Then $\tau$ is indeed a topology on $\Z$.
Proof
Recall the definition of the Furstenberg topology:
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 $\Z$.
$\Box$
In view of Union from Synthetic Basis is Topology it suffices to show that $\BB$ is a synthetic basis on $\Z$.
Recall the definition of synthetic basis:
A synthetic basis on $S$ is a subset $\BB \subseteq \powerset S$ of the power set of $S$ such that:
| \((\text B 1)\) | $:$ | $\BB$ is a cover for $S$ | |||||||
| \((\text B 2)\) | $:$ | \(\ds \forall U, V \in \BB:\) | $\exists \AA \subseteq \BB: U \cap V = \bigcup \AA$ |
That is, the intersection of any pair of elements of $\BB$ is a union of sets of $\BB$.
$(\text B 1)$
$\BB$ is trivially a cover of $\Z$, since $\Z \in \BB$.
$\Box$
$(\text B 2)$
Let $a_1 \Z + b_1, a_2 \Z + b_2 \in \BB$.
If:
- $\paren {a_1 \Z + b_1} \cap \paren {a_2 \Z + b_2} = \O$
then it is done, since $\O = \bigcup \O$ and $\O \subseteq \BB$.
Now, suppose that:
- $\exists x \in \paren {a_1 \Z + b_1} \cap \paren {a_2 \Z + b_2}$
Let $\lcm \set {a_1, a_2}$ be the lowest common multiple of $a_1$ and $a_2$.
Then:
| \(\ds y\) | \(\in\) | \(\ds \paren {a_1 \Z + b_1} \cap \paren {a_2 \Z + b_2}\) | ||||||||||||
| \(\ds \leadstoandfrom \ \ \) | \(\ds \forall i \in \set {1,2}: \, \) | \(\ds y\) | \(\in\) | \(\ds a_i \Z + b_i\) | Definition of Set Intersection | |||||||||
| \(\ds \leadstoandfrom \ \ \) | \(\ds \forall i \in \set {1,2}: \, \) | \(\ds y - x\) | \(\in\) | \(\ds a_i \Z + b_i - x\) | ||||||||||
| \(\ds \leadstoandfrom \ \ \) | \(\ds \forall i \in \set {1,2}: \, \) | \(\ds y - x\) | \(\in\) | \(\ds a_i \Z\) | as $b_i - x \in a_i \Z$ | |||||||||
| \(\ds \leadstoandfrom \ \ \) | \(\ds y - x\) | \(\in\) | \(\ds a_1 \Z \cap a_2 \Z\) | Definition of Set Intersection | ||||||||||
| \(\ds \leadstoandfrom \ \ \) | \(\ds y - x\) | \(\in\) | \(\ds \lcm \set {a_1, a_2} \Z\) | Intersection of Integer Ideals is Lowest Common Multiple | ||||||||||
| \(\ds \leadstoandfrom \ \ \) | \(\ds y\) | \(\in\) | \(\ds \lcm \set {a_1, a_2} \Z + x\) |
That is:
- $\paren {a_1 \Z + b_1} \cap \paren {a_2 \Z + b_2} = \lcm \set {a_1, a_2} \Z + x$
This concludes the proof, because:
- $\lcm \set {a_1, a_2} \Z + x = \bigcup \AA$
where:
- $\AA := \set {\lcm \set {a_1, a_2} \Z + x} \subseteq \BB$
$\blacksquare$