Definition:Extended Long Line
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Definition
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:
- $0, \openint 0 1, 1, \openint 0 1, 2, \openint 0 1, \ldots, \openint 0 1, \alpha, \openint 0 1, \alpha + 1, \openint 0 1, \ldots, \openint 0 1, \Omega - 1, \openint 0 1, \Omega$
Also see
Sources
- 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.): Part $\text {II}$: Counterexamples: $46$. The Extended Long Line