Definition:Long Line

Definition

Let $\Omega$ denote the first uncountable ordinal.

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

Let $L$ be the set constructed as follows.

Between each ordinal $\alpha \in \hointr 0 \Omega$ 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 long line.

Informally it can be seen that $L$ is of the form:

$0, \openint 0 1, 1, \openint 0 1, 2, \openint 0 1, \ldots, \openint 0 1, \alpha, \openint 0 1, \alpha + 1, \openint 0 1, \ldots, \openint 0 1, \Omega - 1, \openint 0 1$

Also see

• Definition:Extended Long Line

Sources

• 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.): Part $\text {II}$: Counterexamples: $45$. The Long Line