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.

$\blacksquare$