Long line

In topology, the long line is a topological space analogous to the real line, only much longer. Because it behaves locally just like the real line, but has different large-scale properties, it serves as one of the basic counterexamples of topology.

Table of contents 1 Definition 2 Properties 3 References 4 Telephone terminology

Definition

We start with the first uncountable ordinal ω₁. This is a totally ordered set, and the cartesian product ω₁ × [0, 1) becomes a totally ordered set if we use the lexicographic or dictionary order[?]. The long line L is defined as ω₁ × [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' end-to-end. Compare this with the real interval [0, ∞), which can be viewed as a countable number of copies of [0, 1) pasted together end-to-end. A related space, the extended long line, L*, is obtained by adjoining an additional element to the end of L.

Properties

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 path-connected and simply connected but not contractible. L is a one-dimensional topological manifold with boundary. L is first countable but not second countable.

The extended long line L* is compact; it is the one-point compactification of L. It is also connected, but not path-connected 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.

References

• Lynn Arthur Steen and J. Arthur Seebach, Jr; Counterexamples in Topology; Springer; ISBN 0-387-90312-7

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Telephone terminology

In telephone systems nomenclature a long line is a transmission line in a long-distance 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 spun-off into the corporate entity known as AT&T Long Distance with the breakup of AT&T.