\begin{definition}[Definition:Extended Long Line]
Let $\Omega$ denote the first uncountable ordinal.
Let $\closedint 0 \Omega$ denote the open ordinal space on $\Omega$.
Let $L^*$ be the set constructed as follows.
Between each ordinal $\alpha \in \closedint 0 \Omega$ and its successor $\alpha + 1$, let a copy of the open (real) unit interval $\openint 0 1$ be inserted.
Let a total ordering $\preccurlyeq$ be applied to $L^*$ according to the betweenness described above.
Let the order topology $\tau$ be applied to the ordered structure $\struct {L^*, \preccurlyeq}$.
The resulting topological space $\struct {L^*, \preccurlyeq, \tau}$ is called the '''extended long line'''.
Informally it can be seen that $L$ is of the form:
:$0, \openint 0 1, 1, \openint 0 1, 2, \openint 0 1, \ldots, \openint 0 1, \alpha, \openint 0 1, \alpha + 1, \openint 0 1, \ldots, \openint 0 1, \Omega - 1, \openint 0 1, \Omega$
\end{definition}