Real Number Line is Second-Countable

Theorem
Let $\left({\R, \tau_d}\right)$ be the real number line under the Euclidean metric considered as a topological space.

Then $\left({\R, \tau_d}\right)$ is second-countable.

It follows immediately that $\left({\R, \tau_d}\right)$ is also first-countable and Lindelöf.

Proof
From Countable Basis of Real Number Space we have that $\left({\R, \tau_d}\right)$ has a countable basis.

The result follows directly from the definition of a second-countable space.

Then we have that a Second-Countable Space is First-Countable and a Second-Countable Space is Lindelöf.