Definition:Euclidean Space/Euclidean Topology/Real Number Line

Definition
Let $\R$ denote the real number line.

Let $d: \R \times \R \to \R$ denote the Euclidean metric on $\R$.

Let $\tau_d$ denote the topology on $\R$ induced by $d$.

The topology $\tau_d$ induced by $d$ is called the Euclidean topology.

Hence $\struct {\R, \tau_d}$ is referred to as the real number line under the Euclidean topology.