Arens-Fort Space is Sigma-Compact

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Theorem

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


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


Proof 1

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 $I_S: \left({S, \tau_d}\right) \to \left({S, \tau}\right)$.

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.

$\blacksquare$


Proof 2

The result follows from Arens-Fort Space is Countable and Countable Space is Sigma-Compact.

$\blacksquare$