Uncountable Discrete Space is not Sigma-Compact

Theorem
Let $T = \left({S, \vartheta}\right)$ be a topological space where $\vartheta$ is the discrete topology on $S$.

Let $S$ be an uncountable set, thereby making $\vartheta$ the uncountable discrete topology on $S$.

Then $T$ is not $\sigma$-compact.

Proof
We have that an Uncountable Discrete Space is Not Lindelöf.

But a $\sigma$-compact space is Lindelöf.

So the uncountable discrete topology can not be $\sigma$-compact.

Also see

 * Countable Discrete Space is Sigma-Compact