Topology induced by Scaled Euclidean Metric on Positive Integers is Discrete

Theorem

Let $\delta: \Z_{>0} \times \Z_{>0} \to \R$ be the metric on $\Z_{>0}$ defined as:

$\forall x, y \in \Z_{>0}: \map \delta {x, y} = \dfrac {\size {x - y} } {x y}$

Let $\tau_\delta$ denote the metric topology for $\delta$.

In Scaled Euclidean Metric is Metric it is demonstrated that $\delta$ is indeed a metric on $\Z_{>0}$.

Let $a \in \Z_{>0}$.

Recall the definition of the open $\epsilon$-ball of $a$ in $\struct {\Z_{>0}, \delta}$:

$\map {B_\epsilon} a := \set {x \in A: \map \delta {x, a} < \epsilon}$

Let $x \in \R_{>0}$.

Let $\epsilon \in \R_{>0}$ such that $\epsilon < \dfrac 1 {a \paren {a + 1} }$.

But we have:

$\ds \forall x \in \Z_{>0}, x \ne a: \, $

$\ds \map \delta {x, a}$

$=$

$\ds \frac {\size {x - a} } {x a}$

$\ds $

$=$

$\ds \size {\frac 1 x - \frac 1 a}$

$\ds $

$\ge$

$\ds \size {\frac 1 {a + 1} - \frac 1 a}$

$\ds $

$=$

$\ds \epsilon$

and so:

$\forall x \in \Z_{>0}, x \ne a: x \notin \map {B_\epsilon} a$

It follows that:

$\map {B_\epsilon} a := \set a$

Thus by definition of $\tau_d$:

$\forall a \in \Z_{>0}: \set a \in \tau_\delta$

It follows from Basis for Discrete Topology that $\struct {\Z_{>0}, \tau_\delta}$ is a discrete topological space.

$\blacksquare$

1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $32$. Special Subsets of the Real Line: $10$