Long Line (topology) - Properties

Properties

The closed long ray L = ω₁ × [0,1) consists of an uncountable number of copies of [0,1) 'pasted together' end-to-end. Compare this with the fact that for any countable ordinal α, pasting together α copies of [0,1) gives a space which is still homeomorphic (and order-isomorphic) to [0,1). (And if we tried to glue together more than ω₁ copies of [0,1), the resulting space would no longer be locally homeomorphic to R.)

Every increasing sequence in L converges to a limit in L; this is a consequence of the facts that (1) the elements of ω₁ are the countable ordinals, (2) the supremum of every countable family of countable ordinals is a countable ordinal, and (3) every increasing and bounded sequence of real numbers converges. Consequently, there can be no strictly increasing function L→R.

As order topologies, the (possibly extended) long rays and lines are normal Hausdorff spaces. All of them have the same cardinality as the real line, yet they are 'much longer'. All of them are locally compact. None of them is metrisable; this can be seen as the long ray is sequentially compact but not compact, or even Lindelöf.

The (non-extended) long line or ray is not paracompact. It is path-connected, locally path-connected and simply connected but not contractible. It is a one-dimensional topological manifold, with boundary in the case of the closed ray. It is first-countable but not second countable and not separable, so authors who require the latter properties in their manifolds do not call the long line a manifold.

The long line or ray can be equipped with the structure of a (non-separable) differentiable manifold (with boundary in the case of the closed ray). However, contrary to the topological structure which is unique (topologically, there is only one way to make the real line "longer" at either end), the differentiable structure is not unique: in fact, for each natural number k there exist infinitely many Ck+1 or C∞ structures on the long line or ray inducing any given Ck structure on it. This is in sharp contrast with the situation for ordinary (that is, separable) manifolds, where a Ck structure uniquely determines a C∞ structure as soon as k≥1.

It makes sense to consider all the long spaces at once because every connected (non-empty) one-dimensional (not necessarily separable) topological manifold possibly with boundary, is homeomorphic to either the circle, the closed interval, the open interval (real line), the half-open interval, the closed long ray, the open long ray, or the long line.

The long line or ray can even be equipped with the structure of a (real) analytic manifold (with boundary in the case of the closed ray). However, this is much more difficult than for the differentiable case (it depends on the classification of (separable) one-dimensional analytic manifolds, which is more difficult than for differentiable manifolds). Again, any given C∞ structure can be extended in infinitely many ways to different Cω (=analytic) structures.

The long line or ray cannot be equipped with a Riemannian metric that induces its topology. The reason is that Riemannian manifolds, even without the assumption of paracompactness, can be shown to be metrizable.

The extended long ray L* is compact. It is the one-point compactification of the closed long ray L, but it is also its Stone-Čech compactification, because any continuous function from the (closed or open) long ray to the real line is eventually constant. L* is also connected, but not path-connected because the long line is 'too long' to be covered by a path, which is a continuous image of an interval. L* is not a manifold and is not first countable.