Real Number Line is not Compact

Theorem
Let $\struct {\R, \tau_d}$ be the real number line with the usual (Euclidean) topology.

Then $\struct {\R, \tau_d}$ is not compact.

Proof
We have:
 * Compact Space is Countably Compact
 * Real Number Line is not Countably Compact

Hence, as $\struct {\R, \tau_d}$ is not countably compact, it follows that it is not compact.