Uncountable Discrete Space is not Sigma-Compact

Theorem
Let $T = \left({S, \tau}\right)$ be an uncountable discrete topological space.

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 an uncountable discrete space can not be $\sigma$-compact.

Also see

 * Countable Discrete Space is Sigma-Compact