\begin{definition}[Definition:Ordinal Space/Open/Uncountable]
Let $\Gamma$ be a limit ordinal.
Let $\Omega$ denote the first uncountable ordinal.
The '''uncountable open ordinal space on $\Omega$''' is the particular case of an open ordinal space $\hointr 0 \Gamma$ where $\Gamma = \Omega$.
That is, it is the set $\hointr 0 \Omega$ of all ordinal numbers (strictly) less than $\Omega$, together with the order topology.
\end{definition}