\begin{definition}[Definition:Euclidean Space/Euclidean Topology/Real Number Line]
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 with the Euclidean topology'''.
\end{definition}