Long line (topology)

From Academic Kids

In topology, the long line is a topological space analogous to the real line, but 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.

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 dictionary or lexicographical 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

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