# Uncountable Open Ordinal Space is not Paracompact

## 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 paracompact space.

## Proof

Aiming for a contradiction, suppose $\hointr 0 \Omega$ is a paracompact space.

From Paracompact Space is Metacompact, it follows that $\hointr 0 \Omega$ is a metacompact space.

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

Hence the result by Proof by Contradiction.

$\blacksquare$