Uncountable Open Ordinal Space is not Sigma-Compact

Theorem
Let $\Omega$ denote the first uncountable ordinal.

Let $\hointr 0 \Omega$ denote the open ordinal space on $\Omega$.

Then $\hointr 0 \Omega$ is not a $\sigma$-compact space.

Proof
$\hointr 0 \Omega$ is a $\sigma$-compact space.

From Sigma-Compact Space is Lindelöf, $\hointr 0 \Omega$ is a Lindelöf space.

But this contradicts the fact that from Uncountable Open Ordinal Space is not Lindelöf, $\hointr 0 \Omega$ is not a Lindelöf space.

Hence the result by Proof by Contradiction.