Real Number Line is Non-Meager

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 of the second category.

Proof
We have that the Real Number Line is Complete Metric Space.

From the Baire Category Theorem, a complete metric space is also a Baire space.

The result follows from Baire Space is Second Category.