Let $\mathbb{R}$ denote the real line with its usual absolute value $|\cdot|$, let $\mathbb{Q}\subset \mathbb{R}$ denote the set of rational numbers, let $\mathbb{Z}\subset \mathbb{Q}$ denote the set of integers, and let $\mathbb{N}=\{1,2,3,\dots\}$. Let $\sqrt{2}\in \mathbb{R}$ denote the unique positive real number satisfying $(\sqrt{2})^2=2$. For integers $p,q\in \mathbb{Z}$, let $\gcd(p,q)$ denote their greatest common divisor. Let $d: \mathbb{Q} \times \mathbb{Q} \to [0,\infty)$ be the metric defined by $d(x,y)=|x-y|$ for all $x,y \in \mathbb{Q}$. Then the [metric space](/page/Metric%20Space) $(\mathbb{Q},d)$ is not complete.