The Euclidean Distance on the Real Line is the Absolute Value Metric

Let $\mathbb{R}$ be the set of real numbers, let $|\cdot|$ be the absolute value on $\mathbb{R}$, let $d_{\mathbb{R}}$ be the metric of The Absolute Value Metric on the Real Line, and let $d_{E}$ be the Euclidean distance on $\mathbb{R}^{n}$ in the case $n=1$, with $\mathbb{R}^{1}$ identified with $\mathbb{R}$.

Then $d_{E}(s,t)=d_{\mathbb{R}}(s,t)$ for all $s,t\in\mathbb{R}$. In particular $d_{E}$ and $d_{\mathbb{R}}$ are the same function, so a subset of $\mathbb{R}$ is open with respect to one exactly when it is open with respect to the other, the two metrics determine the same topology by Metric Open Sets Form a Topology, and a subset of $\mathbb{R}$ is compact with respect to one exactly when it is compact with respect to the other.