It behaves locally just like the real line, but has different large-scale properties (e.g., it is neither Lindelöf nor separable).
laid end-to-end, whereas the long line is constructed from an uncountable number of such segments.
is defined as the Cartesian product of the first uncountable ordinal
The open long ray is obtained from the closed long ray by removing the smallest element
More rigorously, it can be defined as the order topology on the disjoint union of the reversed open long ray (“reversed” means the order is reversed) (this is the negative half) and the (not reversed) closed long ray (the positive half), totally ordered by letting the points of the latter be greater than the points of the former.
of the other, and define the long line to be the topological space obtained by gluing the two open long rays along the open interval identified between the two.
(The former construction is better in the sense that it defines the order on the long line and shows that the topology is the order topology; the latter is better in the sense that it uses gluing along an open set, which is clearer from the topological point of view.)
In many uses or counterexamples, however, the distinction is unessential, because the important part is the “long” end of the line, and it doesn't matter what happens at the other end (whether long, short, or closed).
A related space, the (closed) extended long ray,
the resulting space would no longer be locally homeomorphic to
As order topologies, the (possibly extended) long rays and lines are normal Hausdorff spaces.
The (non-extended) long line or ray is not paracompact.
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.
[2] 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.
[3] 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, there are uncountably many (
to be precise) pairwise non-diffeomorphic smooth structures on it.
[4] This is in sharp contrast to the real line, where there are also different smooth structures, but all of them are diffeomorphic to the standard one.
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).
(=analytic) structures (which are pairwise non-diffeomorphic as analytic manifolds).
[5] 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.
It is the one-point compactification of the closed long ray
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.
There exists a p-adic analog of the long line, which is due to George Bergman.
[8] This space is constructed as the increasing union of an uncountable directed set of copies
of the ring of p-adic integers, indexed by a countable ordinal
isomorphism classes of non-paracompact surfaces, even when a generalization of paracompactness, ω-boundedness, is assumed.