Arens-Fort Space is Sigma-Compact

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

Then $T$ is a $\sigma$-compact space.

Proof
The set $S$ is countable because is product of countable sets. Also the identity function $Id:(S,\tau_d)\to (S,\tau)$ where the topology $\tau_d$ is the discrete topology. This function is continuous using Mapping from Discrete Topology is Continuous.

From Countable Discrete Space is Sigma-Compact and Sigma-Compactness Preserved under Continuous Surjection we conclude that $T$ is a $\sigma$-compact space.