The Distance on the Real Numbers

In the setting of The Real Numbers, with the Natural Numbers, Integers and Rationals Identified with Subsets of the Reals, and Completeness, $|s-t|\in\mathbb{R}$ for all $s,t\in\mathbb{R}$, so the map below exists and is unique by Sets and Maps: Ordinary Notation §maps.

¶ The distance on $\mathbb{R}$ is the map $d_{\mathbb{R}}:\mathbb{R}\times\mathbb{R}\to\mathbb{R}$ with

$$d_{\mathbb{R}}(s,t)=|s-t|\qquad\text{for all }s,t\in\mathbb{R}.$$