Topology induced by Usual Metric on Positive Integers is Discrete

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Theorem

Let $\Z_{>0}$ be the set of (strictly) positive integers.

Let $d: \Z_{>0} \times \Z_{>0} \to \R$ be the usual (Euclidean) metric on $\Z_{>0}$.


Then the metric topology for $d$ is a discrete topology.


Proof

Let $\tau_d$ denote the metric topology for $d$.

Let $\epsilon \in \R_{>0}$ such that $\epsilon < 1$.

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

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

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

But we have:

$\forall x \in \Z_{>0}, x \ne a: \map d {x, a} \ge 1$

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 $\tau_d$:

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

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

$\blacksquare$


Sources

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