Definition:Real Number Line with Euclidean Topology

Definition
Let $\R$ be the set of real numbers.

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

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

Then $\left({\R, \tau_d}\right)$ is the real number space.

Also see

 * Real Number Space is Topological Space