Arens-Fort Space is Sigma-Compact/Proof 1

Theorem

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

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

Proof

From Arens-Fort Space is Countable we have that $S$ is countably infinite.

Let $\tau_d$ be the discrete topology on $S$.

Consider the identity function $I_S: \struct {S, \tau_d} \to \struct {S, \tau}$.

From Mapping from Discrete Space is Continuous, $I_S$ is continuous.

From Identity Mapping is Surjection, $I_S$ is a surjection.

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

$\blacksquare$