\begin{definition}[Definition:Altered Long Line]
Let $\struct {L, \preccurlyeq, \tau}$ denote the '''long line'''.
Let $p$ be a point which is not an element of $L$.
Consider the set $S := L \cup \set p$.
Let $T = \struct {S, \tau_p}$ be the topological space whose topology $\tau_p$ is defined as follows:
The open sets of $T$ are the open sets of $L$ together with those generated by the following neighborhoods of $P$:
:$\ds \map {U_s} p := \set p \cup \set {\bigcup_{\alpha \mathop = \beta}^\Omega \openint \alpha {\alpha + 1}: 1 \le \beta < \Omega}$
where $\Omega$ denotes the first uncountable ordinal.
$\map {U_s} p$ is therefore a right-hand ray without the ordinals.
The topological space $T = \struct {S, \tau_p}$ is known as the '''altered long line'''.
\end{definition}