Arens-Fort Space is Countable

Theorem
Let $T = \left({S, \tau}\right)$ be the Arens-Fort space.

Then $S$ is countably infinite.

Proof
From the definition of the Arens-Fort space:

$S$ is the the Cartesian product $\Z_+ \times \Z_+$ of the set of all positive integers:
 * $S = \left\{{0, 1, 2, \ldots}\right\} \times \left\{{0, 1, 2, \ldots}\right\}$

We have by definition that $\Z_+$ is countable.

From Cartesian Product of Countable Sets it follows that $S = \Z_+ \times \Z_+$ is likewise countable.