
We start with the first uncountable ordinal ω_{1}. This is a totally ordered set, and the cartesian product ω_{1} × [0, 1) becomes a totally ordered set if we use the lexicographic or dictionary order[?]. The long line L is defined as ω_{1} × [0, 1) with the order topology arising from this total order. That is, it consists of an uncountable number of copies of [0, 1) 'pasted together' endtoend. Compare this with the real interval [0, ∞), which can be viewed as a countable number of copies of [0, 1) pasted together endtoend. A related space, the extended long line, L*, is obtained by adjoining an additional element to the end of L.
Both L and L* are normal Hausdorff spaces because they are order topologies. Both of them have the same cardinality as the real line, yet they are 'much longer'. Both of them are locally compact. Neither of them is metrisable.
The long line L is not paracompact. It is pathconnected and simply connected but not contractible. L is a onedimensional topological manifold with boundary. L is first countable but not second countable.
The extended long line L* is compact; it is the onepoint compactification of L. It is also connected, but not pathconnected because the long line is 'too long' to be covered by a path, which is an image of an interval. L* is not a manifold and is not first countable.
In telephone systems nomenclature a long line is a transmission line in a longdistance communications network such as carrier systems, microwave radio links, geosynchronous satellite links, underground cables, aerial cables and open wire, and submarine cables. In the United States, some of this technology was spunoff into the corporate entity known as AT&T Long Distance with the breakup of AT&T.
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