Arens-Fort Space is Countable
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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.
$\blacksquare$