# 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 $\Z_{\ge 0} \times \Z_{\ge 0}$ 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 $\Z_{\ge 0}$ is countable.

From Cartesian Product of Countable Sets is Countable it follows that $S = \Z_{\ge 0} \times \Z_{\ge 0}$ is likewise countable.

$\blacksquare$